science solve the following problems

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Biology library

Course: biology library   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation..

• Observation: the toaster won't toast.

2. Ask a question.

• Question: Why won't my toaster toast?

3. Propose a hypothesis.

• Hypothesis: Maybe the outlet is broken.

4. Make predictions.

• Prediction: If I plug the toaster into a different outlet, then it will toast the bread.

5. Test the predictions.

• Test of prediction: Plug the toaster into a different outlet and try again.
• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• Iteration time!
• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

• Science Notes Posts
• Contact Science Notes
• Todd Helmenstine Biography
• Anne Helmenstine Biography
• Free Printable Periodic Tables (PDF and PNG)
• Periodic Table Wallpapers
• Interactive Periodic Table
• Periodic Table Posters
• How to Grow Crystals
• Chemistry Projects
• Fire and Flames Projects
• Holiday Science
• Chemistry Problems With Answers
• Physics Problems
• Unit Conversion Example Problems
• Chemistry Worksheets
• Biology Worksheets
• Periodic Table Worksheets
• Physical Science Worksheets
• Science Lab Worksheets
• My Amazon Books

Potential And Kinetic Energy Example Problem – Work and Energy Examples

Potential energy is energy attributed to an object by virtue of its position. When the position is changed, the total energy remains unchanged but some potential energy gets converted into kinetic energy . The frictionless roller coaster is a classic potential and kinetic energy example problem.

The roller coaster problem shows how to use the conservation of energy to find the velocity or position or a cart on a frictionless track with different heights. The total energy of the cart is expressed as a sum of its gravitational potential energy and kinetic energy. This total energy remains constant across the length of the track.

Potential And Kinetic Energy Example Problem

A cart travels along a frictionless roller coaster track. At point A, the cart is 10 m above the ground and traveling at 2 m/s. A) What is the velocity at point B when the cart reaches the ground? B) What is the velocity of the cart at point C when the cart reaches a height of 3 m? C) What is the maximum height the cart can reach before the cart stops?

The total energy of the cart is expressed by the sum of its potential energy and its kinetic energy.

Potential energy of an object in a gravitational field is expressed by the formula

where PE is the potential energy m is the mass of the object g is the acceleration due to gravity = 9.8 m/s 2 h is the height above the measured surface.

Kinetic energy is the energy of the object in motion. It is expressed by the formula

KE = ½mv 2

where KE is the kinetic energy m is the mass of the object v is the velocity of the object.

The total energy of the system is conserved at any point of the system. The total energy is the sum of the potential energy and the kinetic energy.

Total E = KE + PE

To find the velocity or position, we need to find this total energy. At point A, we know both the velocity and the position of the cart.

Total E = KE + PE Total E = ½mv 2  + mgh Total E = ½m(2 m/s) 2  + m(9.8 m/s 2 )(10 m) Total E = ½m(4 m 2 /s 2 ) + m(98 m 2 /s 2 ) Total E = m(2 m 2 /s 2 ) + m(98 m 2 /s 2 ) Total E = m(100 m 2 /s 2 )

We can leave the mass value as it appears for now. As we complete each part, you will see what happens to this variable.

The cart is at ground level at point B, so h = 0 m.

Total E = ½mv 2  + mgh Total E = ½mv 2  + mg(0 m) Total E = ½mv 2

All of the energy at this point is kinetic energy. Since total energy is conserved, the total energy at point B is the same as the total energy at point A.

Total E at A = Total Energy at B m(100 m 2 /s 2 ) = ½mv 2

Divide both sides by m 100 m 2 /s 2 = ½v 2

Multiply both sides by 2 200 m 2 /s 2 = v 2

v = 14.1 m/s

The velocity at point B is 14.1 m/s.

At point C, we know only a value for h (h = 3 m).

Total E = ½mv 2 + mgh Total E = ½mv 2 + mg(3 m)

As before, the total energy is conserved. Total energy at A = total energy at C.

m(100 m 2 /s 2 ) = ½mv 2 + m(9.8 m/s 2 )(3 m) m(100 m 2 /s 2 ) = ½mv 2 + m(29.4 m 2 /s 2 )

Divide both sides by m

100 m 2 /s 2 = ½v 2 + 29.4 m 2 /s 2 ½v 2 = (100 – 29.4) m 2 /s 2 ½v 2 = 70.6 m 2 /s 2 v 2 = 141.2 m 2 /s 2 v = 11.9 m/s

The velocity at point C is 11.9 m/s.

The cart will reach its maximum height when the cart stops or v = 0 m/s.

Total E = ½mv 2 + mgh Total E = ½m(0 m/s) 2 + mgh Total E = mgh

Since total energy is conserved, the total energy at point A is the same as the total energy at point D.

m(100 m 2 /s 2 ) = mgh

100 m 2 /s 2  = gh

100 m 2 /s 2  = (9.8 m/s 2 ) h

The maximum height of the cart is 10.2 m.

A) The velocity of the cart at ground level is 14.1 m/s. B) The velocity of the cart at a height of 3 m is 11.9 m/s. C) The maximum height of the cart is 10.2 m.

This type of problem has one main key point: total energy is conserved at all points of the system. If you know the total energy at one point, you know the total energy at all points.

Related Posts

6.1 Solving Problems with Newton’s Laws

Learning objectives.

By the end of this section, you will be able to:

• Apply problem-solving techniques to solve for quantities in more complex systems of forces
• Use concepts from kinematics to solve problems using Newton’s laws of motion
• Solve more complex equilibrium problems
• Solve more complex acceleration problems
• Apply calculus to more advanced dynamics problems

Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton’s laws in Newton’s Laws of Motion ; in this chapter, we continue to discuss these strategies and apply a step-by-step process.

Problem-Solving Strategies

We follow here the basics of problem solving presented earlier in this text, but we emphasize specific strategies that are useful in applying Newton’s laws of motion . Once you identify the physical principles involved in the problem and determine that they include Newton’s laws of motion, you can apply these steps to find a solution. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, so the following techniques should reinforce skills you have already begun to develop.

Problem-Solving Strategy

Applying newton’s laws of motion.

• Identify the physical principles involved by listing the givens and the quantities to be calculated.
• Sketch the situation, using arrows to represent all forces.
• Determine the system of interest. The result is a free-body diagram that is essential to solving the problem.
• Apply Newton’s second law to solve the problem. If necessary, apply appropriate kinematic equations from the chapter on motion along a straight line.
• Check the solution to see whether it is reasonable.

Let’s apply this problem-solving strategy to the challenge of lifting a grand piano into a second-story apartment. Once we have determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation. Such a sketch is shown in Figure 6.2 (a). Then, as in Figure 6.2 (b), we can represent all forces with arrows. Whenever sufficient information exists, it is best to label these arrows carefully and make the length and direction of each correspond to the represented force.

As with most problems, we next need to identify what needs to be determined and what is known or can be inferred from the problem as stated, that is, make a list of knowns and unknowns. It is particularly crucial to identify the system of interest, since Newton’s second law involves only external forces. We can then determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 6.2 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated in Newton’s Laws of Motion , the system of interest depends on the question we need to answer. Only forces are shown in free-body diagrams, not acceleration or velocity. We have drawn several free-body diagrams in previous worked examples. Figure 6.2 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Once a free-body diagram is drawn, we apply Newton’s second law. This is done in Figure 6.2 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then the forces can be handled algebraically. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. We do this by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Generally, just write Newton’s second law in components along the different directions. Then, you have the following equations:

(If, for example, the system is accelerating horizontally, then you can then set a y = 0 . a y = 0 . ) We need this information to determine unknown forces acting on a system.

As always, we must check the solution. In some cases, it is easy to tell whether the solution is reasonable. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving; with experience, it becomes progressively easier to judge whether an answer is reasonable. Another way to check a solution is to check the units. If we are solving for force and end up with units of millimeters per second, then we have made a mistake.

There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. We look first at problems involving particle equilibrium, which make use of Newton’s first law, and then consider particle acceleration, which involves Newton’s second law.

Particle Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration, but it is important to remember that these conditions are relative. For example, an object may be at rest when viewed from our frame of reference, but the same object would appear to be in motion when viewed by someone moving at a constant velocity. We now make use of the knowledge attained in Newton’s Laws of Motion , regarding the different types of forces and the use of free-body diagrams, to solve additional problems in particle equilibrium .

Example 6.1

Different tensions at different angles.

Thus, as you might expect,

This gives us the following relationship:

Note that T 1 T 1 and T 2 T 2 are not equal in this case because the angles on either side are not equal. It is reasonable that T 2 T 2 ends up being greater than T 1 T 1 because it is exerted more vertically than T 1 . T 1 .

Now consider the force components along the vertical or y -axis:

This implies

Substituting the expressions for the vertical components gives

There are two unknowns in this equation, but substituting the expression for T 2 T 2 in terms of T 1 T 1 reduces this to one equation with one unknown:

which yields

Solving this last equation gives the magnitude of T 1 T 1 to be

Finally, we find the magnitude of T 2 T 2 by using the relationship between them, T 2 = 1.225 T 1 T 2 = 1.225 T 1 , found above. Thus we obtain

Significance

Particle acceleration.

We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.

Example 6.2

Drag force on a barge.

The drag of the water F → D F → D is in the direction opposite to the direction of motion of the boat; this force thus works against F → app , F → app , as shown in the free-body diagram in Figure 6.4 (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Because the applied forces are perpendicular, the x - and y -axes are in the same direction as F → 1 F → 1 and F → 2 . F → 2 . The problem quickly becomes a one-dimensional problem along the direction of F → app F → app , since friction is in the direction opposite to F → app . F → app . Our strategy is to find the magnitude and direction of the net applied force F → app F → app and then apply Newton’s second law to solve for the drag force F → D . F → D .

The angle is given by

From Newton’s first law, we know this is the same direction as the acceleration. We also know that F → D F → D is in the opposite direction of F → app , F → app , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as F → app , F → app , but its magnitude is slightly less than F → app . F → app . The problem is now one-dimensional. From the free-body diagram, we can see that

However, Newton’s second law states that

This can be solved for the magnitude of the drag force of the water F D F D in terms of known quantities:

Substituting known values gives

The direction of F → D F → D has already been determined to be in the direction opposite to F → app , F → app , or at an angle of 53 ° 53 ° south of west.

In Newton’s Laws of Motion , we discussed the normal force , which is a contact force that acts normal to the surface so that an object does not have an acceleration perpendicular to the surface. The bathroom scale is an excellent example of a normal force acting on a body. It provides a quantitative reading of how much it must push upward to support the weight of an object. But can you predict what you would see on the dial of a bathroom scale if you stood on it during an elevator ride? Will you see a value greater than your weight when the elevator starts up? What about when the elevator moves upward at a constant speed? Take a guess before reading the next example.

Example 6.3

What does the bathroom scale read in an elevator.

From the free-body diagram, we see that F → net = F → s − w → , F → net = F → s − w → , so we have

Solving for F s F s gives us an equation with only one unknown:

or, because w = m g , w = m g , simply

No assumptions were made about the acceleration, so this solution should be valid for a variety of accelerations in addition to those in this situation. ( Note: We are considering the case when the elevator is accelerating upward. If the elevator is accelerating downward, Newton’s second law becomes F s − w = − m a . F s − w = − m a . )

• We have a = 1.20 m/s 2 , a = 1.20 m/s 2 , so that F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) yielding F s = 825 N . F s = 825 N .
• Now, what happens when the elevator reaches a constant upward velocity? Will the scale still read more than his weight? For any constant velocity—up, down, or stationary—acceleration is zero because a = Δ v Δ t a = Δ v Δ t and Δ v = 0 . Δ v = 0 . Thus, F s = m a + m g = 0 + m g F s = m a + m g = 0 + m g or F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , which gives F s = 735 N . F s = 735 N .

Thus, the scale reading in the elevator is greater than his 735-N (165-lb.) weight. This means that the scale is pushing up on the person with a force greater than his weight, as it must in order to accelerate him upward. Clearly, the greater the acceleration of the elevator, the greater the scale reading, consistent with what you feel in rapidly accelerating versus slowly accelerating elevators. In Figure 6.5 (b), the scale reading is 735 N, which equals the person’s weight. This is the case whenever the elevator has a constant velocity—moving up, moving down, or stationary.

Check Your Understanding 6.1

Now calculate the scale reading when the elevator accelerates downward at a rate of 1.20 m/s 2 . 1.20 m/s 2 .

The solution to the previous example also applies to an elevator accelerating downward, as mentioned. When an elevator accelerates downward, a is negative, and the scale reading is less than the weight of the person. If a constant downward velocity is reached, the scale reading again becomes equal to the person’s weight. If the elevator is in free fall and accelerating downward at g , then the scale reading is zero and the person appears to be weightless.

Example 6.4

Two attached blocks.

For block 1: T → + w → 1 + N → = m 1 a → 1 T → + w → 1 + N → = m 1 a → 1

For block 2: T → + w → 2 = m 2 a → 2 . T → + w → 2 = m 2 a → 2 .

Notice that T → T → is the same for both blocks. Since the string and the pulley have negligible mass, and since there is no friction in the pulley, the tension is the same throughout the string. We can now write component equations for each block. All forces are either horizontal or vertical, so we can use the same horizontal/vertical coordinate system for both objects

When block 1 moves to the right, block 2 travels an equal distance downward; thus, a 1 x = − a 2 y . a 1 x = − a 2 y . Writing the common acceleration of the blocks as a = a 1 x = − a 2 y , a = a 1 x = − a 2 y , we now have

From these two equations, we can express a and T in terms of the masses m 1 and m 2 , and g : m 1 and m 2 , and g :

Check Your Understanding 6.2

Calculate the acceleration of the system, and the tension in the string, when the masses are m 1 = 5.00 kg m 1 = 5.00 kg and m 2 = 3.00 kg . m 2 = 3.00 kg .

Example 6.5

Atwood machine.

• We have For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . (The negative sign in front of m 2 a m 2 a indicates that m 2 m 2 accelerates downward; both blocks accelerate at the same rate, but in opposite directions.) Solve the two equations simultaneously (subtract them) and the result is ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . Solving for a : a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 . a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 .
• Observing the first block, we see that T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N . T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N .

Check Your Understanding 6.3

Determine a general formula in terms of m 1 , m 2 m 1 , m 2 and g for calculating the tension in the string for the Atwood machine shown above.

Newton’s Laws of Motion and Kinematics

Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics , and hence the relevance of earlier chapters.

When approaching problems that involve various types of forces, acceleration, velocity, and/or position, listing the givens and the quantities to be calculated will allow you to identify the principles involved. Then, you can refer to the chapters that deal with a particular topic and solve the problem using strategies outlined in the text. The following worked example illustrates how the problem-solving strategy given earlier in this chapter, as well as strategies presented in other chapters, is applied to an integrated concept problem.

Example 6.6

What force must a soccer player exert to reach top speed.

• We are given the initial and final velocities (zero and 8.00 m/s forward); thus, the change in velocity is Δ v = 8.00 m/s Δ v = 8.00 m/s . We are given the elapsed time, so Δ t = 2.50 s . Δ t = 2.50 s . The unknown is acceleration, which can be found from its definition: a = Δ v Δ t . a = Δ v Δ t . Substituting the known values yields a = 8.00 m/s 2.50 s = 3.20 m/s 2 . a = 8.00 m/s 2.50 s = 3.20 m/s 2 .
• Here we are asked to find the average force the ground exerts on the runner to produce this acceleration. (Remember that we are dealing with the force or forces acting on the object of interest.) This is the reaction force to that exerted by the player backward against the ground, by Newton’s third law. Neglecting air resistance, this would be equal in magnitude to the net external force on the player, since this force causes her acceleration. Since we now know the player’s acceleration and are given her mass, we can use Newton’s second law to find the force exerted. That is, F net = m a . F net = m a . Substituting the known values of m and a gives F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N . F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N .

This is a reasonable result: The acceleration is attainable for an athlete in good condition. The force is about 50 pounds, a reasonable average force.

Check Your Understanding 6.4

The soccer player stops after completing the play described above, but now notices that the ball is in position to be stolen. If she now experiences a force of 126 N to attempt to steal the ball, which is 2.00 m away from her, how long will it take her to get to the ball?

Example 6.7

What force acts on a model helicopter.

The magnitude of the force is now easily found:

Check Your Understanding 6.5

Find the direction of the resultant for the 1.50-kg model helicopter.

Example 6.8

Baggage tractor.

• ∑ F x = m system a x ∑ F x = m system a x and ∑ F x = 820.0 t , ∑ F x = 820.0 t , so 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . Since acceleration is a function of time, we can determine the velocity of the tractor by using a = d v d t a = d v d t with the initial condition that v 0 = 0 v 0 = 0 at t = 0 . t = 0 . We integrate from t = 0 t = 0 to t = 3 : t = 3 : d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s . d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s .
• Refer to the free-body diagram in Figure 6.8 (b). ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N . ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N .

Recall that v = d s d t v = d s d t and a = d v d t a = d v d t . If acceleration is a function of time, we can use the calculus forms developed in Motion Along a Straight Line , as shown in this example. However, sometimes acceleration is a function of displacement. In this case, we can derive an important result from these calculus relations. Solving for dt in each, we have d t = d s v d t = d s v and d t = d v a . d t = d v a . Now, equating these expressions, we have d s v = d v a . d s v = d v a . We can rearrange this to obtain a d s = v d v . a d s = v d v .

Example 6.9

Motion of a projectile fired vertically.

The acceleration depends on v and is therefore variable. Since a = f ( v ) , a = f ( v ) , we can relate a to v using the rearrangement described above,

We replace ds with dy because we are dealing with the vertical direction,

We now separate the variables ( v ’s and dv ’s on one side; dy on the other):

Thus, h = 114 m . h = 114 m .

Check Your Understanding 6.6

If atmospheric resistance is neglected, find the maximum height for the mortar shell. Is calculus required for this solution?

Interactive

Explore the forces at work in this simulation when you try to push a filing cabinet. Create an applied force and see the resulting frictional force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. View a free-body diagram of all the forces (including gravitational and normal forces).

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

• Authors: William Moebs, Samuel J. Ling, Jeff Sanny
• Publisher/website: OpenStax
• Book title: University Physics Volume 1
• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws

© Jan 19, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

January 6, 2020

The Most Important Scientific Problems Have Yet to Be Solved

If certain areas of science appear to be quite mature, others are in the process of development, and yet others remain to be born

By Santiago Ramón y Cajal

Santiago Ramón y Cajal in the 1880s.

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

Santiago Ramón y Cajal (1852-1934) was a neuroscientist and pathologist, and Spain’s first Nobel laureate. This excerpt from his book Advice for a Young Investigator was first posted on the MIT Press Reader on January 6, 2020. An essay on his remarkable scientific drawings appeared in Scientific American in 2015.

Here is a false concept often heard from the lips of the newly graduated: “Everything of major importance in the various areas of science has already been clarified. What difference does it make if I add some minor detail or gather up what is left in some field where more diligent observers have already collected the abundant, ripe grain. Science won’t change its perspective because of my work, and my name will never emerge from obscurity.”

This is often indolence masquerading as modesty. However, it is also expressed by worthy young men reflecting on the first pangs of dismay experienced when undertaking some major project. This superficial concept of science must be eradicated by the young investigator who does not wish to fail, hopelessly overcome by the struggle developing in his mind between the utilitarian suggestions that are part and parcel of his ethical environment (which may soon convert him to an ordinary and financially successful general practitioner), and those nobler impulses of duty and loyalty urging him on to achievement and honor.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing . By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

Wanting to earn the trust placed in him by his mentors, the inexperienced observer hopes to discover a new lode at the earth’s surface, where easy exploration will build his reputation quickly. Unfortunately, with his first excursions into the literature hardly begun, he is shocked to find that the metal lies deep within the ground—surface deposits have been virtually exhausted by observers fortunate enough to arrive earlier and exercise their simple right of eminent domain.

It is nevertheless true that if we arrived on the scene too late for certain problems, we were also born too early to help solve others. Within a century we shall come, by the natural course of events, to monopolize science, plunder its major assets, and harvest its vast fields of data.

Yet we must recognize that there are times when, on the heels of a chance discovery or the development of an important new technique, magnificent scientific discoveries occur one after another as if by spontaneous generation. This happened during the Renaissance when Descartes, Pascal, Galileo, Bacon, Boyle, Newton, our own Sanchez, and others revealed clearly the errors of the ancients and spread the belief that the Greeks, far from exhausting the field of science, had scarcely taken the first steps in understanding the universe. It is a wonderful and fortunate thing for a scientist to be born during one of these great decisive moments in the history of ideas, when much of what has been done in the past is invalidated. Under these circumstances, it could not be easier to choose a fertile area of investigation.

However, let us not exaggerate the importance of such events. Instead, bear in mind that even in our own time science is often built on the ruins of theories once thought to be indestructible. It is important to realize that if certain areas of science appear to be quite mature, others are in the process of development, and yet others remain to be born. Especially in biology, where immense amounts of work have been carried out during the last century, the most essential problems remain unsolved—the origin of life, the problems of heredity and development, the structure and chemical composition of the cell, and so on.

It is fair to say that, in general, no problems have been exhausted; instead, men have been exhausted by the problems. Soil that appears impoverished to one researcher reveals its fertility to another. Fresh talent approaching the analysis of a problem without prejudice will always see new possibilities—some aspect not considered by those who believe that a subject is fully understood. Our knowledge is so fragmentary that unexpected findings appear in even the most fully explored topics. Who, a few short years ago, would have suspected that light and heat still held scientific secrets in reserve? Nevertheless, we now have  argon  in the atmosphere, the  x-rays  of Roentgen, and the  radium  of the Curies, all of which illustrate the inadequacy of our former methods, and the prematurity of our former syntheses.

The best application of the following beautiful dictum of Geoffroy Saint-Hilaire is in biology: “The infinite is always before us.” And the same applies to Carnoy’s no less graphic thought: “Science is a perpetual creative process.” Not everyone is destined to venture into the forest and by sheer determination carve out a serviceable road. However, even the most humble among us can take advantage of the path opened by genius and by traveling along it extract one or another secret from the unknown.

If the beginner is willing to accept the role of gathering details that escaped the wise discoverer, he can be assured that those searching for minutiae eventually acquire an analytical sense so discriminating, and powers of observation so keen, that they are able to solve important problems successfully.

So many apparently trivial observations have led investigators with a thorough knowledge of methods to great scientific conquests! Furthermore, we must bear in mind that because science relentlessly differentiates, the minutiae of today often become important principles tomorrow.

It is also essential to remember that our appreciation of what is important and what is minor, what is great and what is small, is based on false wisdom, on a true anthropomorphic error. Superior and inferior do not exist in nature, nor do primary and secondary relationships. The hierarchies that our minds take pleasure in assigning to natural phenomena arise from the fact that instead of considering things individually, and how they are interrelated, we view them strictly from the perspective of their usefulness or the pleasure they give us. In the chain of life all links are equally valuable because all prove equally necessary.

Things that we see from a distance or do not know how to evaluate are considered small. Even assuming the perspective of human egotism, think how many issues of profound importance to humanity lie within the protoplasm of the simplest microbe! Nothing seems more important in bacteriology than a knowledge of infectious bacteria, and nothing more secondary than the inoffensive microbes that grow abundantly in decomposing organic material. Nevertheless, if these humble fungi—whose mission is to return to the general circulation of matter those substances incorporated by the higher plants and animals—were to disappear, humans could not inhabit the planet.

The far-reaching importance of attention to detail in technical methodology is perhaps demonstrated more clearly in biology than in any other sphere. To cite but one example, recall that Koch, the great German bacteriologist, thought of adding a little alkali to a basic aniline dye, and this allowed him to stain and thus discover the tubercle bacillus—revealing the etiology of a disease that had until then remained uncontrolled by the wisdom of the most illustrious pathologists.

Even the most prominent of the great geniuses have demonstrated a lack of intellectual perspective in the appraisal of scientific insights. Today, we can find many seeds of great discoveries that were mentioned as curiosities of little importance in the writings of the ancients, and even in those of the wise men of the Renaissance. Lost in the pages of a confused theological treatise ( Christianismi restitutio ) are three apparently disdainful lines written by Servetus referring to the pulmonary circulation, which now constitute his major claim to fame. The Aragonese philosopher would be surprised indeed if he were to rise from the dead today. He would find his laborious metaphysical disquisitions totally forgotten, whereas the observation he used simply to argue for the residence of the soul in the blood is widely praised! Or again, it has been inferred from a passage of Seneca’s that the ancients knew the magnifying powers of a crystal sphere filled with water. Who would have suspected that in this phenomenon of magnification, disregarded for centuries, slumbered the embryo of two powerful analytical instruments, the microscope and telescope—and two equally great sciences, biology and astronomy!

In summary, there are no small problems. Problems that appear small are large problems that are not understood. Instead of tiny details unworthy of the intellectual, we have men whose tiny intellects cannot rise to penetrate the infinitesimal. Nature is a harmonious mechanism where all parts, including those appearing to play a secondary role, cooperate in the functional whole. In contemplating this mechanism, shallow men arbitrarily divide its parts into essential and secondary, whereas the insightful thinker is content with classifying them as understood and poorly understood, ignoring for the moment their size and immediately useful properties. No one can predict their importance in the future.

How it works

For Business

Join Mind Tools

Article • 5 min read

Using the Scientific Method to Solve Problems

How the scientific method and reasoning can help simplify processes and solve problems.

By the Mind Tools Content Team

The processes of problem-solving and decision-making can be complicated and drawn out. In this article we look at how the scientific method, along with deductive and inductive reasoning can help simplify these processes.

‘It is a capital mistake to theorize before one has information. Insensibly one begins to twist facts to suit our theories, instead of theories to suit facts.’ Sherlock Holmes

The Scientific Method

The scientific method is a process used to explore observations and answer questions. Originally used by scientists looking to prove new theories, its use has spread into many other areas, including that of problem-solving and decision-making.

The scientific method is designed to eliminate the influences of bias, prejudice and personal beliefs when testing a hypothesis or theory. It has developed alongside science itself, with origins going back to the 13th century. The scientific method is generally described as a series of steps.

• observations/theory
• explanation/conclusion

The first step is to develop a theory about the particular area of interest. A theory, in the context of logic or problem-solving, is a conjecture or speculation about something that is not necessarily fact, often based on a series of observations.

Once a theory has been devised, it can be questioned and refined into more specific hypotheses that can be tested. The hypotheses are potential explanations for the theory.

The testing, and subsequent analysis, of these hypotheses will eventually lead to a conclus ion which can prove or disprove the original theory.

Applying the Scientific Method to Problem-Solving

How can the scientific method be used to solve a problem, such as the color printer is not working?

1. Use observations to develop a theory.

In order to solve the problem, it must first be clear what the problem is. Observations made about the problem should be used to develop a theory. In this particular problem the theory might be that the color printer has run out of ink. This theory is developed as the result of observing the increasingly faded output from the printer.

2. Form a hypothesis.

Note down all the possible reasons for the problem. In this situation they might include:

• The printer is set up as the default printer for all 40 people in the department and so is used more frequently than necessary.
• There has been increased usage of the printer due to non-work related printing.
• In an attempt to reduce costs, poor quality ink cartridges with limited amounts of ink in them have been purchased.
• The printer is faulty.

All these possible reasons are hypotheses.

3. Test the hypothesis.

Once as many hypotheses (or reasons) as possible have been thought of, then each one can be tested to discern if it is the cause of the problem. An appropriate test needs to be devised for each hypothesis. For example, it is fairly quick to ask everyone to check the default settings of the printer on each PC, or to check if the cartridge supplier has changed.

4. Analyze the test results.

Once all the hypotheses have been tested, the results can be analyzed. The type and depth of analysis will be dependant on each individual problem, and the tests appropriate to it. In many cases the analysis will be a very quick thought process. In others, where considerable information has been collated, a more structured approach, such as the use of graphs, tables or spreadsheets, may be required.

5. Draw a conclusion.

Based on the results of the tests, a conclusion can then be drawn about exactly what is causing the problem. The appropriate remedial action can then be taken, such as asking everyone to amend their default print settings, or changing the cartridge supplier.

Inductive and Deductive Reasoning

The scientific method involves the use of two basic types of reasoning, inductive and deductive.

Inductive reasoning makes a conclusion based on a set of empirical results. Empirical results are the product of the collection of evidence from observations. For example:

‘Every time it rains the pavement gets wet, therefore rain must be water’.

There has been no scientific determination in the hypothesis that rain is water, it is purely based on observation. The formation of a hypothesis in this manner is sometimes referred to as an educated guess. An educated guess, whilst not based on hard facts, must still be plausible, and consistent with what we already know, in order to present a reasonable argument.

Deductive reasoning can be thought of most simply in terms of ‘If A and B, then C’. For example:

• if the window is above the desk, and
• the desk is above the floor, then
• the window must be above the floor

It works by building on a series of conclusions, which results in one final answer.

Social Sciences and the Scientific Method

The scientific method can be used to address any situation or problem where a theory can be developed. Although more often associated with natural sciences, it can also be used to develop theories in social sciences (such as psychology, sociology and linguistics), using both quantitative and qualitative methods.

Quantitative information is information that can be measured, and tends to focus on numbers and frequencies. Typically quantitative information might be gathered by experiments, questionnaires or psychometric tests. Qualitative information, on the other hand, is based on information describing meaning, such as human behavior, and the reasons behind it. Qualitative information is gathered by way of interviews and case studies, which are possibly not as statistically accurate as quantitative methods, but provide a more in-depth and rich description.

The resultant information can then be used to prove, or disprove, a hypothesis. Using a mix of quantitative and qualitative information is more likely to produce a rounded result based on the factual, quantitative information enriched and backed up by actual experience and qualitative information.

In terms of problem-solving or decision-making, for example, the qualitative information is that gained by looking at the ‘how’ and ‘why’ , whereas quantitative information would come from the ‘where’, ‘what’ and ‘when’.

It may seem easy to come up with a brilliant idea, or to suspect what the cause of a problem may be. However things can get more complicated when the idea needs to be evaluated, or when there may be more than one potential cause of a problem. In these situations, the use of the scientific method, and its associated reasoning, can help the user come to a decision, or reach a solution, secure in the knowledge that all options have been considered.

Join Mind Tools and get access to exclusive content.

This resource is only available to Mind Tools members.

Already a member? Please Login here

Get 20% off your first year of Mind Tools

Our on-demand e-learning resources let you learn at your own pace, fitting seamlessly into your busy workday. Join today and save with our limited time offer!

Sign-up to our newsletter

Subscribing to the Mind Tools newsletter will keep you up-to-date with our latest updates and newest resources.

Subscribe now

Business Skills

Personal Development

Leadership and Management

Member Extras

Most Popular

Newest Releases

Pain Points Podcast - Balancing Work And Kids

Pain Points Podcast - Improving Culture

Mind Tools Store

About Mind Tools Content

Discover something new today

Pain points podcast - what is ai.

Exploring Artificial Intelligence

Pain Points Podcast - How Do I Get Organized?

It's Time to Get Yourself Sorted!

How Emotionally Intelligent Are You?

Boosting Your People Skills

Self-Assessment

What's Your Leadership Style?

Learn About the Strengths and Weaknesses of the Way You Like to Lead

Study Guides > College Algebra CoRequisite Course

Using formulas to solve problems, learning outcomes.

• Set up a linear equation involving distance, rate, and time.
• Find the dimensions of a rectangle given the area.
• Find the dimensions of a box given information about its side lengths.

Recall the relationship between distance, rate and time

Example: solving an application using a formula, analysis of the solution, how were the fractions handled in the example above.

• In the text of the solution when solving for [latex]r[/latex] the first time, the parentheses were eliminated using the distributive property, then operations on fractions were used to combine like terms.

[latex]\begin{array}{c}r\left(\frac{1}{2}\right)=\left(r - 10\right)\left(\frac{2}{3}\right)\hfill \\ \frac{1}{2}r=\frac{2}{3}r-\frac{20}{3}\hfill \\ \frac{1}{2}r-\frac{2}{3}r=-\frac{20}{3}\hfill \\ -\frac{1}{6}r=-\frac{20}{3}\hfill \\ r=-\frac{20}{3}\left(-6\right)\hfill \\ r=40\hfill \end{array}[/latex]

• Later, in the analysis of the solution, the LCD between the denominators [latex]2 \text{ and } 3 [/latex] was multiplied on both sides of the equation to cancel out the denominators so that operations on fractions were not necessary. Do you see how the LCD wasn't actually multiplied through on both sides, but that the denominators cancelled out, resulting in a linear equation in one variable without denominators?

[latex]\begin{array}{l}r\left(\frac{1}{2}\right)=\left(r - 10\right)\left(\frac{2}{3}\right)\hfill \\ 6\times r\left(\frac{1}{2}\right)=6\times \left(r - 10\right)\left(\frac{2}{3}\right)\hfill \\ 3r=4\left(r - 10\right)\hfill \\ 3r=4r - 40\hfill \\ -r=-40\hfill \\ r=40\hfill \end{array}[/latex]

How To: Solve Multi-Step Equations 1. (Optional) Multiply to clear any fractions or decimals.

2. Simplify each side by clearing parentheses and combining like terms.

3. Add or subtract to isolate the variable term—you may have to move a term with the variable.

4. Multiply or divide to isolate the variable.

5. Check the solution.

Answer: 45 [latex]\frac{\text{mi}}{\text{h}}[/latex]

Example: Solving a Perimeter Problem

Evaluating a variable for an expression

[latex]\begin{array}{l}P=2L+2W\hfill \\ 54=2\left(L\right)+2W \,\,\,\,\,\,\,\, \text{ wrap the } L \text{ to see where to make the substitution} \hfill \\ 54=2\left(W+3\right)+2W \,\,\,\,\, \text{ drop the expression for } L \text{ into the parentheses}\hfill \\ 54=2W+6+2W\hfill \\ 54=4W+6\end{array}[/latex]

Answer: L = 37 cm, W = 18 cm

Example: Solving an Area Problem

Answer: The standard formula for area is [latex]A=LW[/latex]; however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one. We know that the length is 6 in. more than the width, so we can write length as [latex]L=W+6[/latex]. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length. [latex]\begin{array}{l}P=2L+2W\hfill \\ 48=2\left(W+6\right)+2W\hfill \\ 48=2W+12+2W\hfill \\ 48=4W+12\hfill \\ 36=4W\hfill \\ 9=W\hfill \\ \left(9+6\right)=L\hfill \\ 15=L\hfill \end{array}[/latex] Now, we find the area given the dimensions of [latex]L=15[/latex] in. and [latex]W=9[/latex] in. [latex]\begin{array}{l}A\hfill&=LW\hfill \\ A\hfill&=15\left(9\right)\hfill \\ \hfill&=135\text{ in}^{2}\hfill \end{array}[/latex] The area is [latex]135[/latex] in 2 .

Answer: 250 ft 2

Example: Solving a Volume Problem

Licenses & attributions, cc licensed content, original.

• Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .

CC licensed content, Shared previously

• College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/ [email protected] .
• Question ID 52436. Authored by: Edward Wicks. License: CC BY: Attribution . License terms: IMathAS Community License CC- BY + GPL.
• Question ID 7679. Authored by: Tyler Wallace. License: CC BY: Attribution . License terms: IMathAS Community License CC- BY + GPL.
• Question ID 1688. Authored by: WebWork-Rochester. License: CC BY: Attribution . License terms: IMathAS Community License CC- BY + GPL.

Please add a message.

Message received. Thanks for the feedback.

• Illustrations
• Problem Sets
• Calculators

You are here

Punnett square practice problems.

• Alleles, genotype and phenotype
• Punnett Square
• Genotype and phenotype probabilities with a monohybrid cross
• Allele, genotype and Phenotype questions
• Monohybrid genotype and phenotype probability questions

Search form

Latest illustrations.

• Ocean Gyres and Geostrophic Flow
• Langmuir Circulation
• Test sensitivity - specificity calculator
• How earthquakes show us the inside of the Earth
• Surface currents, the Ekman spiral, and Ekman transport

© Andrew Staroscik 2011-2022

• Bipolar Disorder
• Therapy Center
• When To See a Therapist
• Types of Therapy
• Best Online Therapy
• Best Couples Therapy
• Best Family Therapy
• Managing Stress
• Sleep and Dreaming
• Understanding Emotions
• Self-Improvement
• Healthy Relationships
• Student Resources
• Personality Types
• Verywell Mind Insights
• 2023 Verywell Mind 25
• Mental Health in the Classroom
• Editorial Process
• Meet Our Review Board
• Crisis Support

Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List

Identifying problems and solutions in scientific text

Kevin heffernan.

Department of Computer Science and Technology, University of Cambridge, 15 JJ Thomson Avenue, Cambridge, CB3 0FD UK

Simone Teufel

Research is often described as a problem-solving activity, and as a result, descriptions of problems and solutions are an essential part of the scientific discourse used to describe research activity. We present an automatic classifier that, given a phrase that may or may not be a description of a scientific problem or a solution, makes a binary decision about problemhood and solutionhood of that phrase. We recast the problem as a supervised machine learning problem, define a set of 15 features correlated with the target categories and use several machine learning algorithms on this task. We also create our own corpus of 2000 positive and negative examples of problems and solutions. We find that we can distinguish problems from non-problems with an accuracy of 82.3%, and solutions from non-solutions with an accuracy of 79.7%. Our three most helpful features for the task are syntactic information (POS tags), document and word embeddings.

Introduction

Problem solving is generally regarded as the most important cognitive activity in everyday and professional contexts (Jonassen 2000 ). Many studies on formalising the cognitive process behind problem-solving exist, for instance (Chandrasekaran 1983 ). Jordan ( 1980 ) argues that we all share knowledge of the thought/action problem-solution process involved in real life, and so our writings will often reflect this order. There is general agreement amongst theorists that state that the nature of the research process can be viewed as a problem-solving activity (Strübing 2007 ; Van Dijk 1980 ; Hutchins 1977 ; Grimes 1975 ).

One of the best-documented problem-solving patterns was established by Winter ( 1968 ). Winter analysed thousands of examples of technical texts, and noted that these texts can largely be described in terms of a four-part pattern consisting of Situation, Problem, Solution and Evaluation. This is very similar to the pattern described by Van Dijk ( 1980 ), which consists of Introduction-Theory, Problem-Experiment-Comment and Conclusion. The difference is that in Winter’s view, a solution only becomes a solution after it has been evaluated positively. Hoey changes Winter’s pattern by introducing the concept of Response in place of Solution (Hoey 2001 ). This seems to describe the situation in science better, where evaluation is mandatory for research solutions to be accepted by the community. In Hoey’s pattern, the Situation (which is generally treated as optional) provides background information; the Problem describes an issue which requires attention; the Response provides a way to deal with the issue, and the Evaluation assesses how effective the response is.

An example of this pattern in the context of the Goldilocks story can be seen in Fig.  1 . In this text, there is a preamble providing the setting of the story (i.e. Goldilocks is lost in the woods), which is called the Situation in Hoey’s system. A Problem in encountered when Goldilocks becomes hungry. Her first Response is to try the porridge in big bear’s bowl, but she gives this a negative Evaluation (“too hot!”) and so the pattern returns to the Problem. This continues in a cyclic fashion until the Problem is finally resolved by Goldilocks giving a particular Response a positive Evaluation of baby bear’s porridge (“it’s just right”).

Example of problem-solving pattern when applied to the Goldilocks story.

Reproduced with permission from Hoey ( 2001 )

It would be attractive to detect problem and solution statements automatically in text. This holds true both from a theoretical and a practical viewpoint. Theoretically, we know that sentiment detection is related to problem-solving activity, because of the perception that “bad” situations are transformed into “better” ones via problem-solving. The exact mechanism of how this can be detected would advance the state of the art in text understanding. In terms of linguistic realisation, problem and solution statements come in many variants and reformulations, often in the form of positive or negated statements about the conditions, results and causes of problem–solution pairs. Detecting and interpreting those would give us a reasonably objective manner to test a system’s understanding capacity. Practically, being able to detect any mention of a problem is a first step towards detecting a paper’s specific research goal. Being able to do this has been a goal for scientific information retrieval for some time, and if successful, it would improve the effectiveness of scientific search immensely. Detecting problem and solution statements of papers would also enable us to compare similar papers and eventually even lead to automatic generation of review articles in a field.

There has been some computational effort on the task of identifying problem-solving patterns in text. However, most of the prior work has not gone beyond the usage of keyword analysis and some simple contextual examination of the pattern. Flowerdew ( 2008 ) presents a corpus-based analysis of lexio-grammatical patterns for problem and solution clauses using articles from professional and student reports. Problem and solution keywords were used to search their corpora, and each occurrence was analysed to determine grammatical usage of the keyword. More interestingly, the causal category associated with each keyword in their context was also analysed. For example, Reason–Result or Means-Purpose were common causal categories found to be associated with problem keywords.

The goal of the work by Scott ( 2001 ) was to determine words which are semantically similar to problem and solution, and to determine how these words are used to signal problem-solution patterns. However, their corpus-based analysis used articles from the Guardian newspaper. Since the domain of newspaper text is very different from that of scientific text, we decided not to consider those keywords associated with problem-solving patterns for use in our work.

Instead of a keyword-based approach, Charles ( 2011 ) used discourse markers to examine how the problem-solution pattern was signalled in text. In particular, they examined how adverbials associated with a result such as “thus, therefore, then, hence” are used to signal a problem-solving pattern.

Problem solving also has been studied in the framework of discourse theories such as Rhetorical Structure Theory (Mann and Thompson 1988 ) and Argumentative Zoning (Teufel et al. 2000 ). Problem- and solutionhood constitute two of the original 23 relations in RST (Mann and Thompson 1988 ). While we concentrate solely on this aspect, RST is a general theory of discourse structure which covers many intentional and informational relations. The relationship to Argumentative Zoning is more complicated. The status of certain statements as problem or solutions is one important dimension in the definitions of AZ categories. AZ additionally models dimensions other than problem-solution hood (such as who a scientific idea belongs to, or which intention the authors might have had in stating a particular negative or positive statement). When forming categories, AZ combines aspects of these dimensions, and “flattens” them out into only 7 categories. In AZ it is crucial who it is that experiences the problems or contributes a solution. For instance, the definition of category “CONTRAST” includes statements that some research runs into problems, but only if that research is previous work (i.e., not if it is the work contributed in the paper itself). Similarly, “BASIS” includes statements of successful problem-solving activities, but only if they are achieved by previous work that the current paper bases itself on. Our definition is simpler in that we are interested only in problem solution structure, not in the other dimensions covered in AZ. Our definition is also more far-reaching than AZ, in that we are interested in all problems mentioned in the text, no matter whose problems they are. Problem-solution recognition can therefore be seen as one aspect of AZ which can be independently modelled as a “service task”. This means that good problem solution structure recognition should theoretically improve AZ recognition.

In this work, we approach the task of identifying problem-solving patterns in scientific text. We choose to use the model of problem-solving described by Hoey ( 2001 ). This pattern comprises four parts: Situation, Problem, Response and Evaluation. The Situation element is considered optional to the pattern, and so our focus centres on the core pattern elements.

Goal statement and task

Many surface features in the text offer themselves up as potential signals for detecting problem-solving patterns in text. However, since Situation is an optional element, we decided to focus on either Problem or Response and Evaluation as signals of the pattern. Moreover, we decide to look for each type in isolation. Our reasons for this are as follows: It is quite rare for an author to introduce a problem without resolving it using some sort of response, and so this is a good starting point in identifying the pattern. There are exceptions to this, as authors will sometimes introduce a problem and then leave it to future work, but overall there should be enough signal in the Problem element to make our method of looking for it in isolation worthwhile. The second signal we look for is the use of Response and Evaluation within the same sentence. Similar to Problem elements, we hypothesise that this formulation is well enough signalled externally to help us in detecting the pattern. For example, consider the following Response and Evaluation: “One solution is to use smoothing”. In this statement, the author is explicitly stating that smoothing is a solution to a problem which must have been mentioned in a prior statement. In scientific text, we often observe that solutions implicitly contain both Response and Evaluation (positive) elements. Therefore, due to these reasons there should be sufficient external signals for the two pattern elements we concentrate on here.

When attempting to find Problem elements in text, we run into the issue that the word “problem” actually has at least two word senses that need to be distinguished. There is a word sense of “problem” that means something which must be undertaken (i.e. task), while another sense is the core sense of the word, something that is problematic and negative. Only the latter sense is aligned with our sense of problemhood. This is because the simple description of a task does not predispose problemhood, just a wish to perform some act. Consider the following examples, where the non-desired word sense is being used:

• “Das and Petrov (2011) also consider the problem of unsupervised bilingual POS induction”. (Chen et al. 2011 ).
• “In this paper, we describe advances on the problem of NER in Arabic Wikipedia”. (Mohit et al. 2012 ).

Here, the author explicitly states that the phrases in orange are problems, they align with our definition of research tasks and not with what we call here ‘problematic problems’. We will now give some examples from our corpus for the desired, core word sense:

• “The major limitation of supervised approaches is that they require annotations for example sentences.” (Poon and Domingos 2009 ).
• “To solve the problem of high dimensionality we use clustering to group the words present in the corpus into much smaller number of clusters”. (Saha et al. 2008 ).

When creating our corpus of positive and negative examples, we took care to select only problem strings that satisfy our definition of problemhood; “ Corpus creation ” section will explain how we did that.

Corpus creation

Our new corpus is a subset of the latest version of the ACL anthology released in March, 2016 1 which contains 22,878 articles in the form of PDFs and OCRed text. 2

The 2016 version was also parsed using ParsCit (Councill et al. 2008 ). ParsCit recognises not only document structure, but also bibliography lists as well as references within running text. A random subset of 2500 papers was collected covering the entire ACL timeline. In order to disregard non-article publications such as introductions to conference proceedings or letters to the editor, only documents containing abstracts were considered. The corpus was preprocessed using tokenisation, lemmatisation and dependency parsing with the Rasp Parser (Briscoe et al. 2006 ).

Definition of ground truth

Our goal was to define a ground truth for problem and solution strings, while covering as wide a range as possible of syntactic variations in which such strings naturally occur. We also want this ground truth to cover phenomena of problem and solution status which are applicable whether or not the problem or solution status is explicitly mentioned in the text.

To simplify the task, we only consider here problem and solution descriptions that are at most one sentence long. In reality, of course, many problem descriptions and solution descriptions go beyond single sentence, and require for instance an entire paragraph. However, we also know that short summaries of problems and solutions are very prevalent in science, and also that these tend to occur in the most prominent places in a paper. This is because scientists are trained to express their contribution and the obstacles possibly hindering their success, in an informative, succinct manner. That is the reason why we can afford to only look for shorter problem and solution descriptions, ignoring those that cross sentence boundaries.

To define our ground truth, we examined the parsed dependencies and looked for a target word (“problem/solution”) in subject position, and then chose its syntactic argument as our candidate problem or solution phrase. To increase the variation, i.e., to find as many different-worded problem and solution descriptions as possible, we additionally used semantically similar words (near-synonyms) of the target words “problem” or “solution” for the search. Semantic similarity was defined as cosine in a deep learning distributional vector space, trained using Word2Vec (Mikolov et al. 2013 ) on 18,753,472 sentences from a biomedical corpus based on all full-text Pubmed articles (McKeown et al. 2016 ). From the 200 words which were semantically closest to “problem”, we manually selected 28 clear synonyms. These are listed in Table  1 . From the 200 semantically closest words to “solution” we similarly chose 19 (Table  2 ). Of the sentences matching our dependency search, a subset of problem and solution candidate sentences were randomly selected.

Selected words for use in problem candidate phrase extraction

Selected words for use in solution candidate phrase extraction

An example of this is shown in Fig.  2 . Here, the target word “drawback” is in subject position (highlighted in red), and its clausal argument (ccomp) is “(that) it achieves low performance” (highlighted in purple). Examples of other arguments we searched for included copula constructions and direct/indirect objects.

Example of our extraction method for problems using dependencies. (Color figure online)

If more than one candidate was found in a sentence, one was chosen at random. Non-grammatical sentences were excluded; these might appear in the corpus as a result of its source being OCRed text.

800 candidates phrases expressing problems and solutions were automatically extracted (1600 total) and then independently checked for correctness by two annotators (the two authors of this paper). Both authors found the task simple and straightforward. Correctness was defined by two criteria:

• An unexplained phenomenon or a problematic state in science; or
• A research question; or
• An artifact that does not fulfil its stated specification.
• The phrase must not lexically give away its status as problem or solution phrase.

The second criterion saves us from machine learning cues that are too obvious. If for instance, the phrase itself contained the words “lack of” or “problematic” or “drawback”, our manual check rejected it, because it would be too easy for the machine learner to learn such cues, at the expense of many other, more generally occurring cues.

Sampling of negative examples

We next needed to find negative examples for both cases. We wanted them not to stand out on the surface as negative examples, so we chose them so as to mimic the obvious characteristics of the positive examples as closely as possible. We call the negative examples ‘non-problems’ and ‘non-solutions’ respectively. We wanted the only differences between problems and non-problems to be of a semantic nature, nothing that could be read off on the surface. We therefore sampled a population of phrases that obey the same statistical distribution as our problem and solution strings while making sure they really are negative examples. We started from sentences not containing any problem/solution words (i.e. those used as target words). From each such sentence, we at random selected one syntactic subtree contained in it. From these, we randomly selected a subset of negative examples of problems and solutions that satisfy the following conditions:

• The distribution of the head POS tags of the negative strings should perfectly match the head POS tags 3 of the positive strings. This has the purpose of achieving the same proportion of surface syntactic constructions as observed in the positive cases.
• The average lengths of the negative strings must be within a tolerance of the average length of their respective positive candidates e.g., non-solutions must have an average length very similar (i.e. + / -  small tolerance) to solutions. We chose a tolerance value of 3 characters.

Again, a human quality check was performed on non-problems and non-solutions. For each candidate non-problem statement, the candidate was accepted if it did not contain a phenomenon, a problematic state, a research question or a non-functioning artefact. If the string expressed a research task, without explicit statement that there was anything problematic about it (i.e., the ‘wrong’ sense of “problem”, as described above), it was allowed as a non-problem. A clause was confirmed as a non-solution if the string did not represent both a response and positive evaluation.

If the annotator found that the sentence had been slightly mis-parsed, but did contain a candidate, they were allowed to move the boundaries for the candidate clause. This resulted in cleaner text, e.g., in the frequent case of coordination, when non-relevant constituents could be removed.

From the set of sentences which passed the quality-test for both independent assessors, 500 instances of positive and negative problems/solutions were randomly chosen (i.e. 2000 instances in total). When checking for correctness we found that most of the automatically extracted phrases which did not pass the quality test for problem-/solution-hood were either due to obvious learning cues or instances where the sense of problem-hood used is relating to tasks (cf. “ Goal statement and task ” section).

Experimental design

In our experiments, we used three classifiers, namely Naïve Bayes, Logistic Regression and a Support Vector Machine. For all classifiers an implementation from the WEKA machine learning library (Hall et al. 2009 ) was chosen. Given that our dataset is small, tenfold cross-validation was used instead of a held out test set. All significance tests were conducted using the (two-tailed) Sign Test (Siegel 1956 ).

Linguistic correlates of problem- and solution-hood

We first define a set of features without taking the phrase’s context into account. This will tell us about the disambiguation ability of the problem/solution description’s semantics alone. In particular, we cut out the rest of the sentence other than the phrase and never use it for classification. This is done for similar reasons to excluding certain ‘give-away’ phrases inside the phrases themselves (as explained above). As the phrases were found using templates, we know that the machine learner would simply pick up on the semantics of the template, which always contains a synonym of “problem” or “solution”, thus drowning out the more hidden features hopefully inherent in the semantics of the phrases themselves. If we allowed the machine learner to use these stronger features, it would suffer in its ability to generalise to the real task.

ngrams Bags of words are traditionally successfully used for classification tasks in NLP, so we included bags of words (lemmas) within the candidate phrases as one of our features (and treat it as a baseline later on). We also include bigrams and trigrams as multi-word combinations can be indicative of problems and solutions e.g., “combinatorial explosion”.

Polarity Our second feature concerns the polarity of each word in the candidate strings. Consider the following example of a problem taken from our dataset: “very conservative approaches to exact and partial string matches overgenerate badly”. In this sentence, words such as “badly” will be associated with negative polarity, therefore being useful in determining problem-hood. Similarly, solutions will often be associated with a positive sentiment e.g. “smoothing is a good way to overcome data sparsity” . To do this, we perform word sense disambiguation of each word using the Lesk algorithm (Lesk 1986 ). The polarity of the resulting synset in SentiWordNet (Baccianella et al. 2010 ) was then looked up and used as a feature.

Syntax Next, a set of syntactic features were defined by using the presence of POS tags in each candidate. This feature could be helpful in finding syntactic patterns in problems and solutions. We were careful not to base the model directly on the head POS tag and the length of each candidate phrase, as these are defining characteristics used for determining the non-problem and non-solution candidate set.

Negation Negation is an important property that can often greatly affect the polarity of a phrase. For example, a phrase containing a keyword pertinent to solution-hood may be a good indicator but with the presence of negation may flip the polarity to problem-hood e.g., “this can’t work as a solution”. Therefore, presence of negation is determined.

Exemplification and contrast Problems and solutions are often found to be coupled with examples as they allow the author to elucidate their point. For instance, consider the following solution: “Once the translations are generated, an obvious solution is to pick the most fluent alternative, e.g., using an n-gram language model”. (Madnani et al. 2012 ). To acknowledge this, we check for presence of exemplification. In addition to examples, problems in particular are often found when contrast is signalled by the author (e.g. “however, “but”), therefore we also check for presence of contrast in the problem and non-problem candidates only.

Discourse Problems and solutions have also been found to have a correlation with discourse properties. For example, problem-solving patterns often occur in the background sections of a paper. The rationale behind this is that the author is conventionally asked to objectively criticise other work in the background (e.g. describing research gaps which motivate the current paper). To take this in account, we examine the context of each string and capture the section header under which it is contained (e.g. Introduction, Future work). In addition, problems and solutions are often found following the Situation element in the problem-solving pattern (cf. “ Introduction ” section). This preamble setting up the problem or solution means that these elements are likely not to be found occurring at the beginning of a section (i.e. it will usually take some sort of introduction to detail how something is problematic and why a solution is needed). Therefore we record the distance from the candidate string to the nearest section header.

Subcategorisation and adverbials Solutions often involve an activity (e.g. a task). We also model the subcategorisation properties of the verbs involved. Our intuition was that since problematic situations are often described as non-actions, then these are more likely to be intransitive. Conversely solutions are often actions and are likely to have at least one argument. This feature was calculated by running the C&C parser (Curran et al. 2007 ) on each sentence. C&C is a supertagger and parser that has access to subcategorisation information. Solutions are also associated with resultative adverbial modification (e.g. “thus, therefore, consequently”) as it expresses the solutionhood relation between the problem and the solution. It has been seen to occur frequently in problem-solving patterns, as studied by Charles ( 2011 ). Therefore, we check for presence of resultative adverbial modification in the solution and non-solution candidate only.

Embeddings We also wanted to add more information using word embeddings. This was done in two different ways. Firstly, we created a Doc2Vec model (Le and Mikolov 2014 ), which was trained on  ∼  19  million sentences from scientific text (no overlap with our data set). An embedding was created for each candidate sentence. Secondly, word embeddings were calculated using the Word2Vec model (cf. “ Corpus creation ” section). For each candidate head, the full word embedding was included as a feature. Lastly, when creating our polarity feature we query SentiWordNet using synsets assigned by the Lesk algorithm. However, not all words are assigned a sense by Lesk, so we need to take care when that happens. In those cases, the distributional semantic similarity of the word is compared to two words with a known polarity, namely “poor” and “excellent”. These particular words have traditionally been consistently good indicators of polarity status in many studies (Turney 2002 ; Mullen and Collier 2004 ). Semantic similarity was defined as cosine similarity on the embeddings of the Word2Vec model (cf. “ Corpus creation ” section).

Modality Responses to problems in scientific writing often express possibility and necessity, and so have a close connection with modality. Modality can be broken into three main categories, as described by Kratzer ( 1991 ), namely epistemic (possibility), deontic (permission / request / wish) and dynamic (expressing ability).

Problems have a strong relationship to modality within scientific writing. Often, this is due to a tactic called “hedging” (Medlock and Briscoe 2007 ) where the author uses speculative language, often using Epistemic modality, in an attempt to make either noncommital or vague statements. This has the effect of allowing the author to distance themselves from the statement, and is often employed when discussing negative or problematic topics. Consider the following example of Epistemic modality from Nakov and Hearst ( 2008 ): “A potential drawback is that it might not work well for low-frequency words”.

To take this linguistic correlate into account as a feature, we replicated a modality classifier as described by (Ruppenhofer and Rehbein 2012 ). More sophisticated modality classifiers have been recently introduced, for instance using a wide range of features and convolutional neural networks, e.g, (Zhou et al. 2015 ; Marasović and Frank 2016 ). However, we wanted to check the effect of a simpler method of modality classification on the final outcome first before investing heavily into their implementation. We trained three classifiers using the subset of features which Ruppenhofer et al. reported as performing best, and evaluated them on the gold standard dataset provided by the authors 4 . The results of the are shown in Table  3 . The dataset contains annotations of English modal verbs on the 535 documents of the first MPQA corpus release (Wiebe et al. 2005 ).

Modality classifier results (precision/recall/f-measure) using Naïve Bayes (NB), logistic regression, and a support vector machine (SVM)

Italicized results reflect highest f-measure reported per modal category

Logistic Regression performed best overall and so this model was chosen for our upcoming experiments. With regards to the optative and concessive modal categories, they can be seen to perform extremely poorly, with the optative category receiving a null score across all three classifiers. This is due to a limitation in the dataset, which is unbalanced and contains very few instances of these two categories. This unbalanced data also is the reason behind our decision of reporting results in terms of recall, precision and f-measure in Table  3 .

The modality classifier was then retrained on the entirety of the dataset used by Ruppenhofer and Rehbein ( 2012 ) using the best performing model from training (Logistic Regression). This new model was then used in the upcoming experiment to predict modality labels for each instance in our dataset.

As can be seen from Table  4 , we are able to achieve good results for distinguishing a problematic statement from non-problematic one. The bag-of-words baseline achieves a very good performance of 71.0% for the Logistic Regression classifier, showing that there is enough signal in the candidate phrases alone to distinguish them much better than random chance.

Results distinguishing problems from non-problems using Naïve Bayes (NB), logistic regression (LR) and a support vector machine (SVM)

Each feature set’s performance is shown in isolation followed by combinations with other features. Tenfold stratified cross-validation was used across all experiments. Statistical significance with respect to the baseline at the p  < 0.05 , 0.01, 0.001 levels is denoted by *, ** and *** respectively

Taking a look at Table  5 , which shows the information gain for the top lemmas,

Information gain (IG) in bits of top lemmas from the bag-of-words baseline in Table  4

we can see that the top lemmas are indeed indicative of problemhood (e.g. “limit”,“explosion”). Bigrams achieved good performance on their own (as did negation and discourse) but unfortunately performance deteriorated when using trigrams, particularly with the SVM and LR. The subcategorisation feature was the worst performing feature in isolation. Upon taking a closer look at our data, we saw that our hypothesis that intransitive verbs are commonly used in problematic statements was true, with over 30% of our problems (153) using them. However, due to our sampling method for the negative cases we also picked up many intransitive verbs (163). This explains the almost random chance performance (i.e.  50%) given that the distribution of intransitive verbs amongst the positive and negative candidates was almost even.

The modality feature was the most expensive to produce, but also didn’t perform very well is isolation. This surprising result may be partly due to a data sparsity issue

where only a small portion (169) of our instances contained modal verbs. The breakdown of how many types of modal senses which occurred is displayed in Table  6 . The most dominant modal sense was epistemic. This is a good indicator of problemhood (e.g. hedging, cf. “ Linguistic correlates of problem- and solution-hood ” section) but if the accumulation of additional data was possible, we think that this feature may have the potential to be much more valuable in determining problemhood. Another reason for the performance may be domain dependence of the classifier since it was trained on text from different domains (e.g. news). Additionally, modality has also shown to be helpful in determining contextual polarity (Wilson et al. 2005 ) and argumentation (Becker et al. 2016 ), so using the output from this modality classifier may also prove useful for further feature engineering taking this into account in future work.

Number of instances of modal senses

Polarity managed to perform well but not as good as we hoped. However, this feature also suffers from a sparsity issue resulting from cases where the Lesk algorithm (Lesk 1986 ) is not able to resolve the synset of the syntactic head.

Knowledge of syntax provides a big improvement with a significant increase over the baseline results from two of the classifiers.

Examining this in greater detail, POS tags with high information gain mostly included tags from open classes (i.e. VB-, JJ-, NN- and RB-). These tags are often more associated with determining polarity status than tags such as prepositions and conjunctions (i.e. adverbs and adjectives are more likely to be describing something with a non-neutral viewpoint).

The embeddings from Doc2Vec allowed us to obtain another significant increase in performance (72.9% with Naïve Bayes) over the baseline and polarity using Word2Vec provided the best individual feature result (77.2% with SVM).

Combining all features together, each classifier managed to achieve a significant result over the baseline with the best result coming from the SVM (81.8%). Problems were also better classified than non-problems as shown in the confusion matrix in Table  7 . The addition of the Word2Vec vectors may be seen as a form of smoothing in cases where previous linguistic features had a sparsity issue i.e., instead of a NULL entry, the embeddings provide some sort of value for each candidate. Particularly wrt. the polarity feature, cases where Lesk was unable to resolve a synset meant that a ZERO entry was added to the vector supplied to the machine learner. Amongst the possible combinations, the best subset of features was found by combining all features with the exception of bigrams, trigrams, subcategorisation and modality. This subset of features managed to improve results in both the Naïve Bayes and SVM classifiers with the highest overall result coming from the SVM (82.3%).

Confusion matrix for problems

The results for disambiguation of solutions from non-solutions can be seen in Table  8 . The bag-of-words baseline performs much better than random, with the performance being quite high with regard to the SVM (this result was also higher than any of the baseline performances from the problem classifiers). As shown in Table  9 , the top ranked lemmas from the best performing model (using information gain) included “use” and “method”. These lemmas are very indicative of solutionhood and so give some insight into the high baseline returned from the machine learners. Subcategorisation and the result adverbials were the two worst performing features. However, the low performance for subcategorisation is due to the sampling of the non-solutions (the same reason for the low performance of the problem transitivity feature). When fitting the POS-tag distribution for the negative samples, we noticed that over 80% of the head POS-tags were verbs (much higher than the problem heads). The most frequent verb type being the infinite form.

Results distinguishing solutions from non-solutions using Naïve Bayes (NB), logistic regression (LR) and a support vector machine (SVM)

Each feature set’s performance is shown in isolation followed by combinations with other features. Tenfold stratified cross-validation was used across all experiments

Information gain (IG) in bits of top lemmas from the bag-of-words baseline in Table  8

This is not surprising given that a very common formulation to describe a solution is to use the infinitive “TO” since it often describes a task e.g., “One solution is to find the singletons and remove them”. Therefore, since the head POS tags of the non-solutions had to match this high distribution of infinitive verbs present in the solution, the subcategorisation feature is not particularly discriminatory. Polarity, negation, exemplification and syntactic features were slightly more discriminate and provided comparable results. However, similar to the problem experiment, the embeddings from Word2Vec and Doc2Vec proved to be the best features, with polarity using Word2Vec providing the best individual result (73.4% with SVM).

Combining all features together managed to improve over each feature in isolation and beat the baseline using all three classifiers. Furthermore, when looking at the confusion matrix in Table  10 the solutions were classified more accurately than the non-solutions. The best subset of features was found by combining all features without adverbial of result, bigrams, exemplification, negation, polarity and subcategorisation. The best result using this subset of features was achieved by the SVM with 79.7%. It managed to greatly improve upon the baseline but was just shy of achieving statistical significance ( p = 0.057 ).

Confusion matrix for solutions

In this work, we have presented new supervised classifiers for the task of identifying problem and solution statements in scientific text. We have also introduced a new corpus for this task and used it for evaluating our classifiers. Great care was taken in constructing the corpus by ensuring that the negative and positive samples were closely matched in terms of syntactic shape. If we had simply selected random subtrees for negative samples without regard for any syntactic similarity with our positive samples, the machine learner may have found easy signals such as sentence length. Additionally, since we did not allow the machine learner to see the surroundings of the candidate string within the sentence, this made our task even harder. Our performance on the corpus shows promise for this task, and proves that there are strong signals for determining both the problem and solution parts of the problem-solving pattern independently.

With regard to classifying problems from non-problems, features such as the POS tag, document and word embeddings provide the best features, with polarity using the Word2Vec embeddings achieving the highest feature performance. The best overall result was achieved using an SVM with a subset of features (82.3%). Classifying solutions from non-solutions also performs well using the embedding features, with the best feature also being polarity using the Word2Vec embeddings, and the highest result also coming from the SVM with a feature subset (79.7%).

In future work, we plan to link problem and solution statements which were found independently during our corpus creation. Given that our classifiers were trained on data solely from the ACL anthology, we also hope to investigate the domain specificity of our classifiers and see how well they can generalise to domains other than ACL (e.g. bioinformatics). Since we took great care at removing the knowledge our classifiers have of the explicit statements of problem and solution (i.e. the classifiers were trained only on the syntactic argument of the explicit statement of problem-/solution-hood), our classifiers should in principle be in a good position to generalise, i.e., find implicit statements too. In future work, we will measure to which degree this is the case.

To facilitate further research on this topic, all code and data used in our experiments can be found here: www.cl.cam.ac.uk/~kh562/identifying-problems-and-solutions.html

Acknowledgements

The first author has been supported by an EPSRC studentship (Award Ref: 1641528). We thank the reviewers for their helpful comments.

1 http://acl-arc.comp.nus.edu.sg/ .

2 The corpus comprises 3,391,198 sentences, 71,149,169 words and 451,996,332 characters.

3 The head POS tags were found using a modification of the Collins’ Head Finder. This modified algorithm addresses some of the limitations of the head finding heuristics described by Collins ( 2003 ) and can be found here: http://nlp.stanford.edu/nlp/javadoc/javanlp/edu/stanford/nlp/trees/ModCollinsHeadFinder.html .

4 https://www.uni-hildesheim.de/ruppenhofer/data/modalia_release1.0.tgz.

Contributor Information

Kevin Heffernan, Email: [email protected] .

Simone Teufel, Email: [email protected] .

• Baccianella S, Esuli A, Sebastiani F. Sentiwordnet 3.0: An enhanced lexical resource for sentiment analysis and opinion mining. LREC. 2010; 10 :2200–2204. [ Google Scholar ]
• Becker, M., Palmer, A., & Frank, A. (2016). Clause types and modality in argumentative microtexts. In Workshop on foundations of the language of argumentation (in conjunction with COMMA) .
• Briscoe, T., Carroll, J., & Watson, R. (2006). The second release of the rasp system. In Proceedings of the COLING/ACL on interactive presentation sessions, association for computational linguistics pp. 77–80.
• Chandrasekaran B. Towards a taxonomy of problem solving types. AI Magazine. 1983; 4 (1):9. [ Google Scholar ]
• Charles M. Adverbials of result: Phraseology and functions in the problem-solution pattern. Journal of English for Academic Purposes. 2011; 10 (1):47–60. doi: 10.1016/j.jeap.2011.01.002. [ CrossRef ] [ Google Scholar ]
• Chen, D., Dyer, C., Cohen, S. B., & Smith, N. A. (2011). Unsupervised bilingual pos tagging with markov random fields. In Proceedings of the first workshop on unsupervised learning in NLP, association for computational linguistics pp. 64–71.
• Collins M. Head-driven statistical models for natural language parsing. Computational Linguistics. 2003; 29 (4):589–637. doi: 10.1162/089120103322753356. [ CrossRef ] [ Google Scholar ]
• Councill, I. G., Giles, C. L., & Kan, M. Y. (2008). Parscit: An open-source CRF reference string parsing package. In LREC .
• Curran, J. R., Clark, S., & Bos, J. (2007). Linguistically motivated large-scale NLP with C&C and boxer. In Proceedings of the 45th annual meeting of the ACL on interactive poster and demonstration sessions, association for computational linguistics pp. 33–36.
• Flowerdew L. Corpus-based analyses of the problem-solution pattern: A phraseological approach. Amsterdam: John Benjamins Publishing; 2008. [ Google Scholar ]
• Grimes JE. The thread of discourse. Berlin: Walter de Gruyter; 1975. [ Google Scholar ]
• Hall M, Frank E, Holmes G, Pfahringer B, Reutemann P, Witten IH. The weka data mining software: An update. ACM SIGKDD Explorations Newsletter. 2009; 11 (1):10–18. doi: 10.1145/1656274.1656278. [ CrossRef ] [ Google Scholar ]
• Hoey M. Textual interaction: An introduction to written discourse analysis. Portland: Psychology Press; 2001. [ Google Scholar ]
• Hutchins J. On the structure of scientific texts. UEA Papers in Linguistics. 1977; 5 (3):18–39. [ Google Scholar ]
• Jonassen DH. Toward a design theory of problem solving. Educational Technology Research and Development. 2000; 48 (4):63–85. doi: 10.1007/BF02300500. [ CrossRef ] [ Google Scholar ]
• Jordan MP. Short texts to explain problem-solution structures-and vice versa. Instructional Science. 1980; 9 (3):221–252. doi: 10.1007/BF00177328. [ CrossRef ] [ Google Scholar ]
• Kratzer, A. (1991). Modality. In von Stechow & Wunderlich (Eds.), Semantics: An international handbook of contemporary research .
• Le QV, Mikolov T. Distributed representations of sentences and documents. ICML. 2014; 14 :1188–1196. [ Google Scholar ]
• Lesk, M. (1986). Automatic sense disambiguation using machine readable dictionaries: How to tell a pine cone from an ice cream cone. In Proceedings of the 5th annual international conference on Systems documentation, ACM (pp. 24–26).
• Madnani, N., Tetreault, J., & Chodorow, M. (2012). Exploring grammatical error correction with not-so-crummy machine translation. In Proceedings of the seventh workshop on building educational applications using NLP, association for computational linguistics pp. 44–53.
• Mann WC, Thompson SA. Rhetorical structure theory: Toward a functional theory of text organization. Text-Interdisciplinary Journal for the Study of Discourse. 1988; 8 (3):243–281. doi: 10.1515/text.1.1988.8.3.243. [ CrossRef ] [ Google Scholar ]
• Marasović, A., & Frank, A. (2016). Multilingual modal sense classification using a convolutional neural network. In Proceedings of the 1st Workshop on Representation Learning for NLP .
• McKeown K, Daume H, Chaturvedi S, Paparrizos J, Thadani K, Barrio P, Biran O, Bothe S, Collins M, Fleischmann KR, et al. Predicting the impact of scientific concepts using full-text features. Journal of the Association for Information Science and Technology. 2016; 67 :2684–2696. doi: 10.1002/asi.23612. [ CrossRef ] [ Google Scholar ]
• Medlock B, Briscoe T. Weakly supervised learning for hedge classification in scientific literature. ACL, Citeseer. 2007; 2007 :992–999. [ Google Scholar ]
• Mikolov, T., Sutskever, I., Chen, K., Corrado, G. S., & Dean, J. (2013). Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems (pp. 3111–3119).
• Mohit, B., Schneider, N., Bhowmick, R., Oflazer, K., & Smith, N. A. (2012). Recall-oriented learning of named entities in arabic wikipedia. In Proceedings of the 13th conference of the European chapter of the association for computational linguistics, association for computational linguistics (pp. 162–173).
• Mullen T, Collier N. Sentiment analysis using support vector machines with diverse information sources. EMNLP. 2004; 4 :412–418. [ Google Scholar ]
• Nakov, P., Hearst, M. A. (2008). Solving relational similarity problems using the web as a corpus. In: ACL (pp. 452–460).
• Poon, H., & Domingos, P. (2009). Unsupervised semantic parsing. In Proceedings of the 2009 conference on empirical methods in natural language processing: Volume 1-association for computational linguistics (pp. 1–10).
• Ruppenhofer, J., & Rehbein, I. (2012). Yes we can!? Annotating the senses of English modal verbs. In Proceedings of the 8th international conference on language resources and evaluation (LREC), Citeseer (pp. 24–26).
• Saha, S. K., Mitra, P., & Sarkar, S. (2008). Word clustering and word selection based feature reduction for maxent based hindi ner. In ACL (pp. 488–495).
• Scott, M. (2001). Mapping key words to problem and solution. In Patterns of text: In honour of Michael Hoey Benjamins, Amsterdam (pp. 109–127).
• Siegel S. Nonparametric statistics for the behavioral sciences. New York: McGraw-hill; 1956. [ Google Scholar ]
• Strübing, J. (2007). Research as pragmatic problem-solving: The pragmatist roots of empirically-grounded theorizing. In The Sage handbook of grounded theory (pp. 580–602).
• Teufel, S., et al. (2000). Argumentative zoning: Information extraction from scientific text. PhD Thesis, Citeseer .
• Turney, P. D. (2002). Thumbs up or thumbs down?: Semantic orientation applied to unsupervised classification of reviews. In Proceedings of the 40th annual meeting on association for computational linguistics, association for computational linguistics (pp. 417–424).
• Van Dijk TA. Text and context explorations in the semantics and pragmatics of discourse. London: Longman; 1980. [ Google Scholar ]
• Wiebe J, Wilson T, Cardie C. Annotating expressions of opinions and emotions in language. Language Resources and Evaluation. 2005; 39 (2):165–210. doi: 10.1007/s10579-005-7880-9. [ CrossRef ] [ Google Scholar ]
• Wilson, T., Wiebe, J., & Hoffmann, P. (2005). Recognizing contextual polarity in phrase-level sentiment analysis. In Proceedings of the conference on human language technology and empirical methods in natural language processing, association for computational linguistics (pp. 347–354).
• Winter, E. O. (1968). Some aspects of cohesion. In Sentence and clause in scientific English . University College London.
• Zhou, M., Frank, A., Friedrich, A., & Palmer, A. (2015). Semantically enriched models for modal sense classification. In Workshop on linking models of lexical, sentential and discourse-level semantics (LSDSem) (p. 44).

Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

Related stories

The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

• Main content

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.12: Scientific Problem Solving

• Last updated
• Save as PDF
• Page ID 52325

How can we use problem solving in our everyday routines?

One day you wake up and realize your clock radio did not turn on to get you out of bed. You are puzzled, so you decide to find out what happened. You list three possible explanations:

• There was a power failure and your radio cannot turn on.
• Your little sister turned it off as a joke.
• You did not set the alarm last night.

Upon investigation, you find that the clock is on, so there is no power failure. Your little sister was spending the night with a friend and could not have turned the alarm off. You notice that the alarm is not set—your forgetfulness made you late. You have used the scientific method to answer a question.

Scientific Problem Solving

Humans have always wondered about the world around them. One of the questions of interest was (and still is): what is this world made of? Chemistry has been defined in various ways as the study of matter. What matter consists of has been a source of debate over the centuries. One of the key areas for this debate in the Western world was Greek philosophy.

The basic approach of the Greek philosophers was to discuss and debate the questions they had about the world. There was no gathering of information to speak of, just talking. As a result, several ideas about matter were put forth, but never resolved. The first philosopher to carry out the gathering of data was Aristotle (384-322 B.C.). He recorded many observations on the weather, on plant and animal life and behavior, on physical motions, and a number of other topics. Aristotle could probably be considered the first "real" scientist, because he made systematic observations of nature and tried to understand what he was seeing.

Inductive and Deductive Reasoning

Two approaches to logical thinking developed over the centuries. These two methods are inductive reasoning and deductive reasoning . Inductive reasoning involves getting a collection of specific examples and drawing a general conclusion from them. Deductive reasoning takes a general principle and then draws a specific conclusion from the general concept. Both are used in the development of scientific ideas.

Inductive reasoning first involves the collection of data: "If I add sodium metal to water, I observe a very violent reaction. Every time I repeat the process, I see the same thing happen." A general conclusion is drawn from these observations: the addition of sodium to water results in a violent reaction.

In deductive reasoning, a specific prediction is made based on a general principle. One general principle is that acids turn blue litmus paper red. Using the deductive reasoning process, one might predict: "If I have a bottle of liquid labeled 'acid', I expect the litmus paper to turn red when I immerse it in the liquid."

The Idea of the Experiment

Inductive reasoning is at the heart of what is now called the " scientific method ." In European culture, this approach was developed mainly by Francis Bacon (1561-1626), a British scholar. He advocated the use of inductive reasoning in every area of life, not just science. The scientific method, as developed by Bacon and others, involves several steps:

• Ask a question - identify the problem to be considered.
• Make observations - gather data that pertains to the question.
• Propose an explanation (a hypothesis) for the observations.
• Make new observations to test the hypothesis further.

Note that this should not be considered a "cookbook" for scientific research. Scientists do not sit down with their daily "to do" list and write down these steps. The steps may not necessarily be followed in order. But this does provide a general idea of how scientific research is usually done.

When a hypothesis is confirmed repeatedly, it eventually becomes a theory—a general principle that is offered to explain natural phenomena. Note a key word— explain , or  explanation . A theory offers a description of why something happens. A law, on the other hand, is a statement that is always true, but offers no explanation as to why. The law of gravity says a rock will fall when dropped, but does not explain why (gravitational theory is very complex and incomplete at present). The kinetic molecular theory of gases, on the other hand, states what happens when a gas is heated in a closed container (the pressure increases), but also explains why (the motions of the gas molecules are increased due to the change in temperature). Theories do not get "promoted" to laws, because laws do not answer the "why" question.

• The early Greek philosophers spent their time talking about nature, but did little or no actual exploration or investigation.
• Inductive reasoning - to develop a general conclusion from a collection of observations.
• Deductive reasoning - to make a specific statement based on a general principle.
• Scientific method - a process of observation, developing a hypothesis, and testing that hypothesis.
• What was the basic shortcoming of the Greek philosophers approach to studying the material world?
• How did Aristotle improve the approach?
• Define “inductive reasoning” and give an example.
• Define “deductive reasoning” and give an example.
• What is the difference between a hypothesis and a theory?
• What is the difference between a theory and a law?

'Third Millennium Thinking': How to use scientific tools to solve everyday problems

Life is full of decisions. “ Third Millennium Thinking: Creating Sense in a World of Knowledge ” outlines methods of making choices rationally using scientific methods.

Nobel Prize-winning physicist Saul Perlmutter , philosophy professor John Campbell , and social psychologist Robert MacCoun turned their course at the University of California Berkeley on using scientific tools to approach everyday problems into a book.

Perlmutter says it’s easy to fall into mental traps or fool ourselves when making a choice. But when people assess all the variables that could influence them and the potential outcomes, they approach questions more thoughtfully, he says.

“There is so much of what is the scientific approach to the world that is never taught anywhere,” Perlmutter says. “It seemed like this was a time for us to be trying to figure out how can we teach this in ways that don’t require having to become a scientist in order to do it.”

Book excerpt: ‘Third Millennium Thinking: Creating Sense in a World of Nonsense’

By Saul Perlmutter, John Campbell and Robert MacCoun

INTRODUCTION

In just the past few decades, those of us who live in the internet-​connected world have obtained access to a nearly unfathomable amount of information. We can click a link and instantly gain insight into whatever we’re curious about, whether it’s treatment options for a particular health condition, how to build a solar generator, or the political history of Malta. On the other hand, sometimes there is so much information we don’t know how to sort or evaluate it. The social science database ProQuest, for example, boasts of “a growing content collection that now encompasses . . . 6 billion digital pages and spans six centuries.” And that’s just old-​school, print information! The Internet Archive’s Wayback Machine, an archive of websites and other digital artifacts dating back to 1996, hosts almost a trillion pages of digital content, tens of millions of books and audios, and nearly a million software programs.

More and more often, it can be hard to determine what to focus on, let alone how to distinguish what’s revelatory and enlightening, in and among all the highly technical, specialized, contradictory, incomplete, out‑of‑date, biased, or deliberately untrue information we can now access. Was that drug study funded by a pharmaceutical company? Did an AI system invent all those supposedly authentic product reviews? What do those statistics leave out? What does that article even mean ? It is also increasingly tricky to identify whom to trust for expert guidance in interpreting this information. There are all sorts of people out there who claim expertise — and perhaps your favorite experts aren’t my favorite experts. Experts disagree, or have ulterior motives, or perhaps don’t understand the world or “real life” beyond their own narrow perspective. How do we find an expert we can safely trust?

To make a sound decision, take a meaningful action, or solve a problem — whether as individuals, in groups, or as a society — we need first to understand reality. But when reality is not easy to discern, and we’re not sure which experts to trust to clarify the matter, we adopt other strategies for navigating the clutter. We “go with our gut”; decide what we “believe” and look for evidence to reaffirm whatever that is; adopt positions based on our affiliations with people we know; even find reassurance in belittling the people who disagree with us. We choose to consult experts who tell us what we like to hear; or bond in shared mistrust of people providing or communicating the information that confuses us, whether they are scientists, scholars, journalists, community leaders, policymakers, or other experts. These coping strategies may help us get by in our personal or professional lives; they may provide a consoling sense of identity or belonging. But they do not actually help us see clearly or make good decisions. And resorting to them can have dangerous social and political consequences.

How can we navigate better — as individuals, and as a society — in this age of informational overwhelm? How do we ward off confusion, avoid mental traps, and sift sense out of nonsense? How do we make decisions and solve problems collaboratively with people who interpret information differently or have different values than we do?

The three of us — a physicist (Saul), a philosopher ( John), and a psychologist (Rob) — have been working closely together for nearly a decade on a project to help our students learn to think about big problems and make effective decisions in this “too much information” age. We began our collaboration in 2011, in response to what was already a worrying trend toward no‑think, politics-​driven decision-​making. An issue like raising the national debt ceiling, for example, was being debated that summer as if it were a religious schism, rather than a simple, practical, probably even testable question of what economic approach would work best to improve the country’s economic well-being. Most of the arguments both yea and nay betrayed equal disregard for, or ignorance of, the most basic principles of scientific thought. We began to wonder whether it might be possible to first articulate and then teach the principles that would lead to clearer thinking, more rational arguments, and a more fruitful collaborative decision-​making process.

The result was a team-​taught, multidisciplinary Big Ideas course at UC Berkeley, intended to teach students the whole gamut of ideas, tools, and approaches that natural and social scientists use to understand the world. We also designed the course to show how useful these approaches can be for everybody in day‑to‑day life, whether working individually or collaboratively, in making reasoned decisions and solving the full range of problems that face us. To our great satisfaction, the course has been both popular and successful, and has since been replicated and adapted by other teachers at a growing number of other universities.1 Our students appear to rethink their worlds and emerge energized with new ways to approach both personal decision-​making and our society’s problems. They are better able to investigate their questions, evaluate information and expertise, and work together as members of a group or a society. Inspired by their enthusiasm, we began to think about new ways to share these tools — and this new way of thinking and working together — beyond the classroom, with students and citizens of all ages.

We have become ever more concerned that our society is losing its way, causing suffering — and missing great opportunities — simply because we don’t have the tools that could help us make sense of the extraordinary amount of complex, often contradictory information now available to us. Practical problem-​solving can come to a standstill when we cannot ascertain the facts of the problems, or, when those problems require communal or political solutions, even agree with others on what those facts are. We humans, who can figure out rocket science and fly to the moon, can’t always figure out how to navigate uncertainty and conflicting points of view to make a simple reasonable decision when we need to.

Part of the problem is that science itself is often a major source of the highly technical, opaque, inconsistent, and contradictory information that has overwhelmed, perplexed, and even angered people. Trust in science has eroded in the recent past.2 The achievements of science cannot live up to all the utopian expectations those successes have generated. Some scientific achievements have also come with negative social, political, or environmental side effects. For these and other reasons, science has become one of the totems of polarization in political discussions. In short, as science became harder to understand, was connected to undesirable side effects, and subjected to politically partisan critiques, many people lost their trust in scientists and in “science” itself.3

But science also has a phenomenal record of providing insight into — if not answers to — the most confounding questions humans have thought to ask. It has helped us to solve puzzles, address problems, and make better lives over millennia. It is a culture of inquiry rooted in the dawn of humankind, with centuries of practice in evaluating conflicting information in a baffling world, and in distinguishing what we know from what we don’t. Along the way, scientists have learned from both successes and mistakes, breakthroughs and blunders, to refine the tools with which to address new questions and solve new problems.

Over the past few years, we have all become aware of the shocking degree of polarization in our society, and the surprising interaction between this polarization and our society’s often-​problematic relationship to science and scientific expertise. If we are to have any hope of finding the practical common plans and common understandings that can move our society ahead together, we need to learn to accept the possibility of errors in our own thinking, and our need for opposing views that help us see where we are going wrong. And we need to understand the source of the disenchantment with and backlash against scientific progress that arose during the end of the Second Millennium and seek to repair it.

No one book and no single approach can heal the rifts. Not all of our polarized disagreements will vanish. But we have to start somewhere. And we believe that one of our more promising starting points is with the culture of science — if we begin to borrow its tools, ideas, and processes, and make a Third Millennium shift in our own thinking.

Adapted from “Third Millenium Thinking” by Saul Perlmutter, John Campbell, and Robert MacCoun. Copyright © 2024 by Saul Perlmutter, John Campbell, and Robert MacCoun. Used with permission of Little, Brown Spark, an imprint of Little, Brown and Company. New York, NY. All rights reserved.

Emiko Tamagawa  produced and edited this interview for broadcast with  Todd Mundt .  Grace Griffin  adapted it for the web.

This article was originally published on WBUR.org.

Copyright 2024 NPR. To see more, visit https://www.npr.org.

• Share full article

For more audio journalism and storytelling, download New York Times Audio , a new iOS app available for news subscribers.

• March 30, 2024   •   9:00 Solving the ‘3 Body Problem’
• March 29, 2024   •   5:26 The Latest on Sean Combs’s Legal Woes
• March 28, 2024   •   9:10 Before Beyoncé: Black Artists Who Crossed Over to Country
• March 27, 2024   •   3:50 A Boxing Novel That Brings the Heat
• March 22, 2024   •   7:47 New Music for Your Weekend
• March 21, 2024   •   6:57 A Surreal New TV Series About Life, Love and Fruit
• March 20, 2024   •   9:37 Curl Up With a Delicious Recipe
• March 15, 2024   •   7:20 Kacey Musgraves Finds Inner Peace on Her New Album
• March 14, 2024   •   5:47 5 Minutes to Love Jazz Flute
• March 12, 2024   •   3:45 Tuxedos Stole the Show at This Year’s Oscars
• March 11, 2024   •   9:13 Oscar Highlights: The Good, the Ken and the Naked
• March 8, 2024   •   6:09 The Alt-Rock Legend Kim Gordon Has a New Album

Solving the ‘3 Body Problem’

Unpacking netflix’s new hit with the times’s cosmic affairs correspondent..

Produced by Alex Barron

Edited by Lynn Levy

Engineered by Efim Shapiro

Featuring Dennis Overbye

The show “3 Body Problem” premiered on March 21 and quickly became one of Netflix’s most-watched titles. It is an adventure story about a group of scientists contending with an extraterrestrial threat. But despite its science fiction trappings, the show is often based in real — and complex — scientific concepts, whether string theory or nanomaterials. In this episode, Dennis Overbye, The Times’s cosmic affairs correspondent, breaks down some of the more brain-bending science behind “3 Body Problem.”

On today’s episode

Dennis Overbye is the cosmic affairs correspondent for The Times, covering physics and astronomy.

The New York Times Audio app is home to journalism and storytelling, and provides news, depth and serendipity. If you haven’t already, download it here — available to Times news subscribers on iOS — and sign up for our weekly newsletter.

Dennis Overbye is the cosmic affairs correspondent for The Times, covering physics and astronomy. More about Dennis Overbye

Advertisement

Kellogg School of Management at Northwestern University

Innovation Data Analytics Sep 10, 2018

How can social science become more solutions-oriented, a conversation between researchers at kellogg and microsoft explores how behavioral science can best be applied..

Noshir Contractor

Duncan Watts

Researchers commonly distinguish between “basic” and “applied” science. Basic science is driven by sheer human curiosity—think knowledge for knowledge’s sake—while applied science draws on this knowledge to solve a specific problem. Engineers who study robotics or nuclear safety, for instance, regularly borrow from the work of physicists, while medical researchers searching for the cure to cancer build from advances in basic biology.

But what happens to a basic discipline without an applied counterpart? Social science aims to investigate how people and societies behave. But who uses its basic principles to understand and solve the many problems facing organizations and societies? And is this fundamentally a problem for the field?

Noshir Contractor , a professor of behavioral sciences at the McCormick School of Engineering, as well as a professor of management and organizations at the Kellogg School and communication studies in the School of Communication at Northwestern, recently sat down with Duncan Watts , principal researcher at Microsoft and an expert in how social influence spreads on networks. They discussed whether social scientists are doing enough to solve problems in the world around us—and what researchers and businesses can do to push the field forward.

This interview has been edited for length and clarity.

Noshir CONTRACTOR: I want to start by asking about an article you wrote last year, which argues that social science should be more practical and solution-oriented. You argued that some of the things that social scientists have held near and dear—like focusing on understanding and explaining phenomena in and of itself—is not going to get us far.

What prompted you to write this article? Why did this become an important issue for you?

Duncan WATTS : Well, the article reflects a frustration I’ve had for almost 20 years.

I come from outside of social science—physics and engineering—so to me the boundaries between economics and political science and psychology and sociology never really made a whole lot of sense. And so it seemed perfectly natural to me to read across all of these different disciplines. When I first started trying to understand social influence and contagion on networks, I started looking for articles that had those words in the titles. And sure enough, I found articles in economics journals, and I found articles in sociology journals and in psychology and political science journals.

And one thing that I found really perplexing and frustrating was that even though they purported to be about the same thing, and would often invoke the same examples, the content was unrecognizably different. It was partly stylistic, but even the mathematics would be impossible to reconcile!

As a scientist you’d like to be able to say, “Well, which mathematical model is better?” But I couldn’t even get to the point where I could express one model in terms of the other. And one makes an assumption that’s fundamentally incommensurable with the other one. So they could both be wrong, but they can’t both be right.

The point I make in the article is that social science has this very theory-oriented perspective on the world. And yet we have this mishmash of theories that don’t really add up. “Organizational behavior” is a perfect example of this. We have hundreds of theories of why organizations do what they do. Yet if you read that literature with the goal of making sense of it, it is really just headache-inducing.

Here’s an example. Take Microsoft, where I work now. A few years ago, Satya Nadella announced a major reorg. This is a multi-hundred billion dollar company. A hundred thousand full-time employees. Tens of thousands of people were moved around. Thousands of people lost their jobs. Thousands of other people got jobs. Everything about the company changed.

We have a hundred years of organization and management science. We have thousands and thousands of papers. You would think that somewhere in that vast volume of things with the word “science” at the end of them, there would be some instructions for Satya Nadella. How should he do this?

I don’t believe there’s an answer to that question in those thousands and thousands of papers. And if that’s not a question that we’re answering in organization science and management science, what are we answering?

CONTRACTOR : So rather than only being motivated by a certain theory, economists and sociologists should try to solve, not just understand, the same problem. And if that becomes the focus, then it follows that they will look anywhere they can for the relevant literature on that problem. That provides the incentive to navigate across disciplines.

But you make an important distinction between purely applied research and the kind you’d like to see more of.

WATTS: Right. What I’m talking about is where you use the applied problem as a way to generate new basic science.

For many years people would approach me after talks about how information spreads in social networks and ask, “How do I get my product to go viral?” And I would say, “That’s not the question we’re asking in sociology. We just kind of inquire about general mechanisms.” But after a while, I thought: maybe we should try to answer that question. That’s not a dumb question, that’s the question that anyone who’s not a social scientist would ask.

You’re trying to address the outsider and say, “Hey, social science is useful. We can actually tell you answers to your questions.” But in order to do that, we’re going to have to generate a lot of basic science ourselves. We don’t have an answer we can pull off the shelf.

“Think of what it would cost to build a theory about how teams interact—a predictive “science of teams.” But if you want to do social science the way physicists do physics, you need your CERNS and LIGOs and Hubble telescopes. ”   — Duncan Watts

CONTRACTOR: This might be a provocative question, but it’s one I’ve given some thought to. When basic science gets applied, we call it engineering. Is part of the issue that social science doesn’t have an equivalent?

Most of us would consider “social engineering” a four-letter word, but computational social science—which uses computational methods to investigate social phenomena—might be a platform for this. How can we use computational social science to address grand societal challenges? How do we accelerate innovation by assembling teams more effectively on the fly? How do we scale up global health solutions when we know the solutions exist, but have not been able to leverage networks well enough to propagate them?

In each of these cases, there’s a very vibrant research agenda. We don’t know enough about how teams are assembled. We don’t know enough about how things spread on the network. So there’s a lot of basic-science questions that are being addressed here. But in the process of addressing them, we’re also showing that we can make a difference. Even if we can’t give the best solution ever, it’s a better solution than the limited solutions we have today.

WATTS: It’s a really interesting point. One answer to the question of why social science isn’t more solution-oriented is that we’re missing this translational bit in the middle, as you say.

I think a second answer is more on the demand side. I wrote a whole book about how people think that social science is obvious, just common sense. Business leaders and politicians think they already know the answers.

Plus, they’re dealing with these incredibly complex problems, but they’re also in a rush. They don’t have time to wait for the research.

And research is expensive. Think of what it would cost to build a theory about how teams interact—a predictive “science of teams.” But if you want to do social science the way physicists do physics, you need your CERNS and LIGOs and Hubble telescopes.

CONTRACTOR: Yes, it would be very costly. But the fact that other areas are able to demand that kind of money says that we are not doing something right in selling our ideas. In other words, people literally don’t value what social science could do. That’s true in terms of funding agencies but also true in terms of businesses.

What do you want others to take away from this conversation?

WATTS: Well, a lot of the data that’s useful to computational social science is owned by industry, and yet a lot of the expertise to make sense of it is in academia. So we need to do more to facilitate industry–academic collaborations. Again, this is not a new idea in engineering. But it’s quite new to the social sciences. If we build collaborations around solving big problems, we can help businesses while also exploring fundamental scientific questions.

CONTRACTOR: The traditional social-scientist model is to go into a company and beg them to be nice enough to say, “Okay, we’ll share some data with you even though you’re not going to come back and help us.” And I think that model is broken. An industry–academic partnership has to be a win–win.

Professor of Management & Organizations; Jane S. & William J. White Professor of Behavioral Sciences, McCormick School of Engineering

The Effects of Climate Change

The effects of human-caused global warming are happening now, are irreversible for people alive today, and will worsen as long as humans add greenhouse gases to the atmosphere.

• We already see effects scientists predicted, such as the loss of sea ice, melting glaciers and ice sheets, sea level rise, and more intense heat waves.
• Scientists predict global temperature increases from human-made greenhouse gases will continue. Severe weather damage will also increase and intensify.

Earth Will Continue to Warm and the Effects Will Be Profound

Global climate change is not a future problem. Changes to Earth’s climate driven by increased human emissions of heat-trapping greenhouse gases are already having widespread effects on the environment: glaciers and ice sheets are shrinking, river and lake ice is breaking up earlier, plant and animal geographic ranges are shifting, and plants and trees are blooming sooner.

Effects that scientists had long predicted would result from global climate change are now occurring, such as sea ice loss, accelerated sea level rise, and longer, more intense heat waves.

The magnitude and rate of climate change and associated risks depend strongly on near-term mitigation and adaptation actions, and projected adverse impacts and related losses and damages escalate with every increment of global warming.

Intergovernmental Panel on Climate Change

Some changes (such as droughts, wildfires, and extreme rainfall) are happening faster than scientists previously assessed. In fact, according to the Intergovernmental Panel on Climate Change (IPCC) — the United Nations body established to assess the science related to climate change — modern humans have never before seen the observed changes in our global climate, and some of these changes are irreversible over the next hundreds to thousands of years.

Scientists have high confidence that global temperatures will continue to rise for many decades, mainly due to greenhouse gases produced by human activities.

The IPCC’s Sixth Assessment report, published in 2021, found that human emissions of heat-trapping gases have already warmed the climate by nearly 2 degrees Fahrenheit (1.1 degrees Celsius) since 1850-1900. 1 The global average temperature is expected to reach or exceed 1.5 degrees C (about 3 degrees F) within the next few decades. These changes will affect all regions of Earth.

The severity of effects caused by climate change will depend on the path of future human activities. More greenhouse gas emissions will lead to more climate extremes and widespread damaging effects across our planet. However, those future effects depend on the total amount of carbon dioxide we emit. So, if we can reduce emissions, we may avoid some of the worst effects.

The scientific evidence is unequivocal: climate change is a threat to human wellbeing and the health of the planet. Any further delay in concerted global action will miss the brief, rapidly closing window to secure a liveable future.

Here are some of the expected effects of global climate change on the United States, according to the Third and Fourth National Climate Assessment Reports:

Future effects of global climate change in the United States:

U.S. Sea Level Likely to Rise 1 to 6.6 Feet by 2100

Global sea level has risen about 8 inches (0.2 meters) since reliable record-keeping began in 1880. By 2100, scientists project that it will rise at least another foot (0.3 meters), but possibly as high as 6.6 feet (2 meters) in a high-emissions scenario. Sea level is rising because of added water from melting land ice and the expansion of seawater as it warms. Image credit: Creative Commons Attribution-Share Alike 4.0

Climate Changes Will Continue Through This Century and Beyond

Global climate is projected to continue warming over this century and beyond. Image credit: Khagani Hasanov, Creative Commons Attribution-Share Alike 3.0

Hurricanes Will Become Stronger and More Intense

Scientists project that hurricane-associated storm intensity and rainfall rates will increase as the climate continues to warm. Image credit: NASA

More Droughts and Heat Waves

Droughts in the Southwest and heat waves (periods of abnormally hot weather lasting days to weeks) are projected to become more intense, and cold waves less intense and less frequent. Image credit: NOAA

Longer Wildfire Season

Warming temperatures have extended and intensified wildfire season in the West, where long-term drought in the region has heightened the risk of fires. Scientists estimate that human-caused climate change has already doubled the area of forest burned in recent decades. By around 2050, the amount of land consumed by wildfires in Western states is projected to further increase by two to six times. Even in traditionally rainy regions like the Southeast, wildfires are projected to increase by about 30%.

Changes in Precipitation Patterns

Climate change is having an uneven effect on precipitation (rain and snow) in the United States, with some locations experiencing increased precipitation and flooding, while others suffer from drought. On average, more winter and spring precipitation is projected for the northern United States, and less for the Southwest, over this century. Image credit: Marvin Nauman/FEMA

Frost-Free Season (and Growing Season) will Lengthen

The length of the frost-free season, and the corresponding growing season, has been increasing since the 1980s, with the largest increases occurring in the western United States. Across the United States, the growing season is projected to continue to lengthen, which will affect ecosystems and agriculture.

Global Temperatures Will Continue to Rise

Summer of 2023 was Earth's hottest summer on record, 0.41 degrees Fahrenheit (F) (0.23 degrees Celsius (C)) warmer than any other summer in NASA’s record and 2.1 degrees F (1.2 C) warmer than the average summer between 1951 and 1980. Image credit: NASA

Arctic Is Very Likely to Become Ice-Free

Sea ice cover in the Arctic Ocean is expected to continue decreasing, and the Arctic Ocean will very likely become essentially ice-free in late summer if current projections hold. This change is expected to occur before mid-century.

U.S. Regional Effects

Climate change is bringing different types of challenges to each region of the country. Some of the current and future impacts are summarized below. These findings are from the Third 3 and Fourth 4 National Climate Assessment Reports, released by the U.S. Global Change Research Program .

• Northeast. Heat waves, heavy downpours, and sea level rise pose increasing challenges to many aspects of life in the Northeast. Infrastructure, agriculture, fisheries, and ecosystems will be increasingly compromised. Farmers can explore new crop options, but these adaptations are not cost- or risk-free. Moreover, adaptive capacity , which varies throughout the region, could be overwhelmed by a changing climate. Many states and cities are beginning to incorporate climate change into their planning.
• Northwest. Changes in the timing of peak flows in rivers and streams are reducing water supplies and worsening competing demands for water. Sea level rise, erosion, flooding, risks to infrastructure, and increasing ocean acidity pose major threats. Increasing wildfire incidence and severity, heat waves, insect outbreaks, and tree diseases are causing widespread forest die-off.
• Southeast. Sea level rise poses widespread and continuing threats to the region’s economy and environment. Extreme heat will affect health, energy, agriculture, and more. Decreased water availability will have economic and environmental impacts.
• Midwest. Extreme heat, heavy downpours, and flooding will affect infrastructure, health, agriculture, forestry, transportation, air and water quality, and more. Climate change will also worsen a range of risks to the Great Lakes.
• Southwest. Climate change has caused increased heat, drought, and insect outbreaks. In turn, these changes have made wildfires more numerous and severe. The warming climate has also caused a decline in water supplies, reduced agricultural yields, and triggered heat-related health impacts in cities. In coastal areas, flooding and erosion are additional concerns.

1. IPCC 2021, Climate Change 2021: The Physical Science Basis , the Working Group I contribution to the Sixth Assessment Report, Cambridge University Press, Cambridge, UK.

2. IPCC, 2013: Summary for Policymakers. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.

3. USGCRP 2014, Third Climate Assessment .

4. USGCRP 2017, Fourth Climate Assessment .

Related Resources

A Degree of Difference

So, the Earth's average temperature has increased about 2 degrees Fahrenheit during the 20th century. What's the big deal?

What’s the difference between climate change and global warming?

“Global warming” refers to the long-term warming of the planet. “Climate change” encompasses global warming, but refers to the broader range of changes that are happening to our planet, including rising sea levels; shrinking mountain glaciers; accelerating ice melt in Greenland, Antarctica and the Arctic; and shifts in flower/plant blooming times.

Is it too late to prevent climate change?

Humans have caused major climate changes to happen already, and we have set in motion more changes still. However, if we stopped emitting greenhouse gases today, the rise in global temperatures would begin to flatten within a few years. Temperatures would then plateau but remain well-elevated for many, many centuries.

Discover More Topics From NASA

Explore Earth Science

Earth Science in Action

Earth Science Data

Facts About Earth

• SUGGESTED TOPICS
• The Magazine
• Newsletters
• Managing Yourself
• Managing Teams
• Work-life Balance
• The Big Idea
• Data & Visuals
• Reading Lists
• Case Selections
• HBR Learning
• Topic Feeds
• Account Settings
• Email Preferences

Is Your AI-First Strategy Causing More Problems Than It’s Solving?

• Oguz A. Acar

Consider a more balanced and thoughtful approach to AI transformation.

The problem with an AI-first strategy lies not within the “AI” but with the notion that it should come “first” aspect. An AI-first approach can be myopic, potentially leading us to overlook the true purpose of technology: to serve and enhance human endeavors. Instead, the author recommends following 3Ps during an AI transformation: problem-centric, people-first, and principle-driven.

From technology giants like Google to major management consultants like McKinsey , a rapidly growing number of companies preach an “AI-first” strategy. In essence, this means considering AI as the ultimate strategic priority , one that precedes other alternative directions. At first glance, this strategy seems logical, perhaps even inevitable. The figures speak for themselves: the sheer volume of investment flowing into AI technologies shows the confidence levels in an increasingly AI-driven future.

• Oguz A. Acar is a Chair in Marketing at King’s Business School, King’s College London.

Partner Center

Follow Polygon online:

• Follow Polygon on Facebook
• Follow Polygon on Youtube
• Follow Polygon on Instagram

Site search

• What to Watch
• What to Play
• PlayStation
• All Entertainment
• Dragon’s Dogma 2
• FF7 Rebirth
• Zelda: Tears of the Kingdom
• Baldur’s Gate 3
• Buyer’s Guides
• Galaxy Brains
• All Podcasts

Filed under:

• Entertainment

The 3-body problem is real, and it’s really unsolvable

Oh god don’t make me explain math

Share this story

• Share this on Facebook
• Share this on Reddit
• Share All sharing options

Share All sharing options for: The 3-body problem is real, and it’s really unsolvable

Everybody seems to be talking about 3 Body Problem , the new Netflix series based on Cixin Liu’s Remembrance of Earth’s Past book trilogy . Fewer people are talking about the two series’ namesake: The unsolvable physics problem of the same name.

This makes sense, because it’s confusing . In physics, the three-body problem attempts to find a way to predict the movements of three objects whose gravity interacts with each of the others — like three stars that are close together in space. Sounds simple enough, right? Yet I myself recently pulled up the Wikipedia article on the three-body problem and closed the tab in the same manner that a person might stagger away from a bright light. Apparently the Earth, sun, and moon are a three-body system? Are you telling me we don’t know how the moon moves ? Scientists have published multiple solutions for the three-body problem? Are you telling me Cixin Liu’s books are out of date?

All I’d wanted to know was why the problem was considered unsolvable, and now memories of my one semester of high school physics were swimming before my eyes like so many glowing doom numbers. However, despite my pains, I have readied several ways that we non-physicists can be confident that the three-body problem is, in fact, unsolvable.

Reason 1: This is a special definition of ‘unsolvable’

The three-body problem is extra confusing, because scientists are seemingly constantly finding new solutions to the three-body problem! They just don’t mean a one-solution-for-all solution. Such a formula does exist for a two-body system, and apparently Isaac Newton figured it out in 1687 . But systems with more than two bodies are, according to physicists, too chaotic (i.e., not in the sense of a child’s messy bedroom, but in the sense of “chaos theory”) to be corralled by a single solution.

When physicists say they have a new solution to the three-body problem, they mean that they’ve found a specific solution for three-body systems that have certain theoretical parameters. Don’t ask me to explain those parameters, because they’re all things like “the three masses are collinear at each instant” or “a zero angular momentum solution with three equal masses moving around a figure-eight shape.” But basically: By narrowing the focus of the problem to certain arrangements of three-body systems, physicists have been able to derive formulas that predict the movements of some of them, like in our solar system. The mass of the Earth and the sun create a “ restricted three-body problem ,” where a less-big body (in this case, the moon) moves under the influence of two massive ones (the Earth and the sun).

What physicists mean when they say the three-body problem has no solution is simply that there isn’t a one-formula-fits-all solution to every way that the gravity of three objects might cause those objects to move — which is exactly what Three-Body Problem bases its whole premise on.

Reason 2: 3 Body Problem picked an unsolved three-body system on purpose

Henri Poincaré’s research into a general solution to the three-body problem formed the basis of what would become known as chaos theory (you might know it from its co-starring role in Jurassic Park ). And 3 Body Problem itself isn’t about any old three-body system. It’s specifically about an extremely chaotic three-body system, the exact kind of arrangement of bodies that Poincaré was focused on when he showed that the problem is “unsolvable.”

[ Ed. note: The rest of this section includes some spoilers for 3 Body Problem .]

In both Liu’s books and Netflix’s 3 Body Problem , humanity faces an invasion by aliens (called Trisolarans in the English translation of the books, and San-Ti in the TV series) whose home solar system features three suns in a chaotic three-body relationship. It is a world where, unlike ours, the heavens are fundamentally unpredictable. Periods of icy cold give way to searing heat that give way to swings in gravity that turn into temporary reprieves that can never be trusted. The unpredictable nature of the San-Ti environment is the source of every detail of their physicality, their philosophy, and their desire to claim Earth for their own.

In other words, 3 Body Problem ’s three-body problem is unsolvable because Liu wanted to write a story with an unsolvable three-body system, so he chose one of the three-body systems for which we have not discovered a solution, and might never.

Reason 3: Scientists are still working on the three-body problem

Perhaps the best reason I can give you to believe that the three-body problem is real, and is really unsolvable, is that some scientists published a whole set of new solutions for specific three-body systems very recently .

If physicists are still working on the three-body problem, we can safely assume that it has not been solved. Scientists, after all, are the real experts. And I am definitely not.

The next level of puzzles.

Take a break from your day by playing a puzzle or two! We’ve got SpellTower, Typeshift, crosswords, and more.

Sign up for the newsletter Patch Notes

A weekly roundup of the best things from Polygon

Just one more thing!

Please check your email to find a confirmation email, and follow the steps to confirm your humanity.

Oops. Something went wrong. Please enter a valid email and try again.

Loading comments...

Don’t have the time to read (or watch) Shōgun? Get the audiobooks for just $10

Cloud’s unreliable narration only makes Final Fantasy 7 Rebirth’s ending more confusing

Imaginary, Lisa Frankenstein, Netflix’s The Beautiful Game, and every new movie to watch at home this weekend

• Dragon’s Dogma 2 guides, walkthroughs, and explainers

Should you give the grimoires to Myrddin or Trysha in Dragon’s Dogma 2?

All maister skills in Dragon’s Dogma 2 and how to get them

How to get a house in FFXIV

Equation Solver

Enter the Equation you want to solve into the editor.

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&