Three TOK essays, marked and unedited.

Can we ever be certain about anything in science and math?

This is a really interesting question that people have been asking for a long time. In my opinion, certainty is possible in math but not really in science, and in this essay I will explain why I think this.

First of all, math is certain because it is based on numbers and numbers do not lie. Everyone agrees that 2 + 2 = 4, and it does not matter what country you are from or what language you speak, it will always be 4. My math teacher always says that math is the only subject where you can be 100% sure that you are right, and I agree with her. When I do a math test and check my answers, I know that if I followed the steps correctly then my answer is correct. You cannot really argue with a math answer like you can with an answer in English or history.

Also, math has been around for thousands of years. The ancient Greeks and Egyptians used math to build things like the pyramids, and the math they used is still the same math that we use today. If math was not certain, then things like bridges and buildings would fall down all the time. The fact that they do not fall down proves that math is certain.

Some people might say that math is not certain because there are some things in math that are confusing, like imaginary numbers. When I first learned about imaginary numbers I thought they made no sense, because how can you have the square root of a negative number? But my teacher explained that they are just another tool that mathematicians use, so this does not really mean that math is uncertain. It just means that some math is hard to understand.

Science is different from math because science changes all the time. Scientists are always discovering new things and changing their minds about old things. For example, when my parents were at school they learned that Pluto was a planet, but now scientists say that Pluto is not a planet anymore. This shows that science is not certain, because something that everyone thought was true turned out to be wrong.

Another example is that people used to think that smoking was not bad for you. In old advertisements, doctors even appeared promoting cigarettes. Now we know that smoking causes cancer and lots of other diseases. So what scientists say can change a lot, and we can never be sure that what they are telling us now is actually true. Maybe in fifty years scientists will say that something we think is healthy now is actually really bad for us.

Also, in science experiments things can go wrong. In my chemistry class we did an experiment to find out the rate of a reaction and everyone in the class got different results, even though we all followed the same method. Some groups made mistakes with measuring and some groups had equipment that was not very accurate. This shows that science cannot be certain, because experiments do not always give the same answer.

However, some people might argue that science can be certain about some things. For example, we know for sure that gravity exists because if you drop something it falls down. Nobody can argue that gravity is not real. But even with gravity, scientists like Newton and Einstein had different ideas about how it works, so we are still not completely certain.

I think the main difference between math and science is that math is in our heads and science is about the real world. The real world is complicated and there are lots of things that can affect it, but math is simple and logical. That is why math can be certain and science cannot.

Other subjects like history are even less certain than science, because historians were not there when things happened and people in the past sometimes lied. For example, we cannot be certain about what really happened in ancient times because there were no cameras. So science is more certain than history but less certain than math. I also think that sometimes people are certain about things just because they feel it, like when you are certain that your friend is telling the truth. But this is not the same as being certain in math.

There is also the question of whether we should even want certainty. If scientists were completely certain about everything, they would stop doing research and we would never discover anything new. So maybe it is good that science is not certain. On the other hand, it would be nice to be certain about things like medicine, because people need to know if a medicine is safe before they take it.

In conclusion, I believe that certainty is attainable in mathematics because numbers are always the same and math answers can be checked. In the natural sciences, certainty is not really attainable because scientists change their minds, experiments can go wrong and the real world is complicated. Overall, math is more certain than science, but both of them are important and we need both of them in our lives.

To what extent is certainty attainable in the natural sciences and mathematics?

Certainty is an important idea in Theory of Knowledge. In this essay, certainty means knowledge that cannot be doubted. Mathematics and the natural sciences are two areas of knowledge that are often seen as the most reliable, so they are good areas to use to explore this question. I will argue that certainty is attainable in mathematics but not in the natural sciences, although both areas give us very reliable knowledge.

Mathematics is based on proofs. A proof starts with axioms, which are statements that are accepted as true, and then uses logical steps to reach a conclusion. Once something has been proven, it stays proven. For example, Pythagoras' theorem says that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. This was proven thousands of years ago and it is still true today. It does not matter who you are or where you live, the theorem will always work. This shows that mathematics can give us certainty.

Another example is the proof that the square root of two is irrational. The proof works by assuming the opposite and showing that this leads to a contradiction. Because the logic is valid, mathematicians are certain that the square root of two cannot be written as a fraction. This kind of reasoning is called deductive reasoning, and it is different from the kind of reasoning used in the sciences.

However, some people argue that mathematics is not completely certain. One reason is that axioms are only assumed to be true. For example, Euclidean geometry is based on axioms that were thought to be obviously true, but in the nineteenth century mathematicians developed non-Euclidean geometries where some of these axioms do not apply. This shows that mathematical knowledge depends on the axioms that are chosen. Another reason is that mathematicians are human and can make mistakes in proofs. Some proofs are so long and complicated that very few people can check them. Also, Kurt Gödel showed that there are limits to what can be proven in mathematics.

Another point is that mathematics is also used to describe the real world. For example, engineers use mathematics to design bridges and scientists use it to make predictions. When mathematics is applied to the real world, the results are not always certain, because the real world is more complicated than the mathematical model. This shows that mathematics is most certain when it stays abstract.

There are also different perspectives on what mathematics is. Some people believe that mathematics is discovered, because mathematical truths exist before humans find them, while others believe that mathematics is invented by humans. If mathematics is invented, it might seem less certain, because humans could have invented it differently. However, even if mathematics is invented, the rules still have to be followed once they are chosen.

Even so, I think mathematics is still certain, because a proof that is valid cannot be wrong. If a mistake is found, the proof was not really a proof in the first place. Non-Euclidean geometry did not show that Euclid was wrong, only that there are other kinds of geometry. So overall mathematics can give us certainty.

The natural sciences are different because they are based on observation and experiment rather than proof. Scientists use the scientific method: they make a hypothesis, test it with experiments, and then draw conclusions. This is inductive reasoning, because it goes from specific observations to general laws. The problem with inductive reasoning is that we can never be sure that the next observation will be the same as the previous ones. For example, people in Europe believed that all swans were white until black swans were found in Australia.

The philosopher Karl Popper argued that scientific theories can never be proven true, they can only be falsified. A good scientific theory makes predictions that could be shown to be false. If the predictions keep being confirmed, we can have more confidence in the theory, but we can never be completely certain. This means that certainty is not attainable in the natural sciences.

A famous example of this is Newtonian physics. Isaac Newton's laws of motion and gravity were accepted for more than two hundred years and they explained a huge number of observations. However, in the early twentieth century Albert Einstein developed the theory of relativity, which showed that Newton's laws do not work in every situation, for example at very high speeds or in very strong gravity. This shows that even a very successful scientific theory can be replaced, so scientific knowledge is not certain.

On the other hand, some scientific knowledge seems certain. For example, it is certain that the Earth orbits the Sun and that water is made of hydrogen and oxygen. Some people would say that it would be unreasonable to doubt these facts. Scientists also use statistics to show how confident they are in their results. However, even these facts are based on evidence, and in theory new evidence could change them. So I still think that certainty is not really attainable in the natural sciences.

Another way scientists try to become more certain is peer review. Before a scientific paper is published, other scientists in the same field read it and check whether the methods and conclusions are sound. Experiments are also repeated by other scientists to see if they get the same results. If many different scientists get the same result, then we can be more confident that the result is correct. However, peer review is done by humans, and sometimes mistakes are not noticed, so this does not make scientific knowledge completely certain either.

When we compare the two areas of knowledge, we can see that they use different methods. Mathematics uses deductive reasoning and proofs, which can give certainty. The natural sciences use inductive reasoning and experiments, which cannot give certainty but can give very reliable knowledge. Both areas of knowledge are important and both help us understand the world in different ways. Scientists also use mathematics in their work, for example when they use equations to describe motion, so the two areas are connected.

In conclusion, certainty is attainable to a large extent in mathematics, because proofs cannot be wrong once they are valid, although mathematics depends on axioms. In the natural sciences, certainty is not attainable, because scientific knowledge is based on evidence and can always change, as the example of Newton and Einstein shows. However, scientific knowledge is still very reliable. Therefore, the answer to the question depends on which area of knowledge we are talking about.

To what extent is certainty attainable in the natural sciences and mathematics?

A claim is certain, in the sense that matters here, when the method used to justify it leaves no room for it to be overturned. That is different from feeling sure. People have felt sure of many things that turned out to be false, so the feeling cannot be the test. The question is whether the methods of mathematics and the natural sciences can produce knowledge that no further evidence or argument could revise. This essay argues that mathematics can reach certainty, but only a conditional kind: certainty that a conclusion follows from its starting points, not that the starting points are true. The natural sciences cannot reach certainty about the world, and their reliability comes from giving up the attempt and measuring their uncertainty instead.

Mathematics seems the strongest candidate because its method is proof. Euclid's proof that there are infinitely many prime numbers is more than two thousand years old and has never needed revision. Suppose there were a largest prime. Multiply all the primes together and add one: the new number is either prime itself or divisible by a prime missing from the list. Either way the supposition collapses. Anyone who follows the steps can check the argument without trusting Euclid, a textbook or an institution, and no experiment could weaken it, because it makes no claim about experience. Here certainty is not a matter of confidence at all. It is a matter of each step being valid.

The history of geometry shows what this certainty covers and what it does not. For about two thousand years Euclid's parallel postulate, which says that through a point outside a line exactly one parallel line can be drawn, looked like a self-evident truth about space. In the nineteenth century Lobachevsky, Bolyai and later Riemann built consistent geometries in which it is false. Einstein's general theory of relativity (1915) then described gravity using curved, non-Euclidean geometry, and observations have supported it. None of Euclid's proofs became invalid. What was lost was the belief that his axioms described physical space. Mathematical certainty survived, but only as certainty of implication: if these axioms hold, these theorems follow. Which axioms describe the world is not something proof can settle.

A stronger objection comes from inside mathematics. In 1931 Kurt Gödel proved that any consistent formal system able to express basic arithmetic contains true statements it cannot prove, and cannot prove its own consistency. Mathematicians therefore cannot be certain, from within such a system, that it will never produce a contradiction. There is a practical limit too. The four colour theorem was proved in 1976 by Appel and Haken with a computer checking well over a thousand cases that no person could check by hand. The proof of the Kepler conjecture that Thomas Hales announced in 1998 was so long that its reviewers said they could not be completely certain it was correct, and a computer-checked formal version was only completed in 2014. In cases like these, mathematical certainty rests partly on trust in machines and in other people.

These objections limit mathematical certainty without removing it. Gödel's theorems were themselves established by proof, so they show the precision of the method as much as its limits: we know exactly which kind of certainty is unavailable. The uncertainty around long computer-assisted proofs concerns human and machine error, not the logic of deduction, and it shrinks as proofs are checked independently. A Platonist, who holds that mathematical objects exist independently of us, would say mathematics gives certain knowledge of that abstract reality. A formalist would say it gives certainty only about what follows from rules we chose. On either view the certainty is internal to mathematics. It is high, but it is conditional.

The natural sciences are in a different position, because their claims are about the world and rest on observation. No number of observations can prove a general law, since the next one might contradict it. Newtonian mechanics is the clearest case. For two centuries it predicted the motion of planets, projectiles and tides with extraordinary success. Yet the orbit of Mercury precessed by about 43 arcseconds per century more than Newton's theory could explain. General relativity accounted for the difference, and observations of starlight during the 1919 solar eclipse supported Einstein's prediction. Newton's laws are still used to plan spacecraft trajectories, so they were not simply wrong, but their claim to be the final description of motion did not survive. If a theory that successful could be revised, success alone cannot deliver certainty.

It could be argued that this sets the bar unreasonably high. Nobody seriously doubts that the Earth orbits the Sun, and treating such claims as uncertain seems perverse. Science also measures its uncertainty precisely. When CERN announced a particle consistent with the Higgs boson in July 2012, the ATLAS and CMS experiments had each reached about five sigma: a random fluctuation of the background at least that strong would be expected roughly once in three and a half million trials. That looks close enough to certainty for any practical purpose.

The OPERA experiment shows why it is not the same thing. In 2011 its team reported neutrinos that appeared to travel faster than light, with a statistical significance of about six sigma, higher than the Higgs announcement. In 2012 the result was traced to a badly connected fibre-optic cable and a timing fault. The statistics had been calculated correctly, but they described only random error, and the error was systematic. Confidence, however carefully measured, is confidence relative to the assumptions built into the measurement. What made the episode a success for science is that the team published the anomaly, asked others to check it and withdrew it when the fault was found. The method works because no result is placed beyond revision.

This is also why the Higgs result deserves more trust than OPERA's, even though its statistical significance was lower. ATLAS and CMS were built by separate teams with different detector designs, so a systematic fault in one would be unlikely to produce the same signal in the other. The confidence placed in the result came less from any single number than from agreement between independent methods, and later measurements of the particle's properties have continued to fit. Scientific knowledge can become so well supported that doubting it would be unreasonable, and a knower is justified in acting on it. That is a kind of practical certainty. But it is a judgment made by a community about the weight of evidence, and it remains open in principle, which is exactly what logical certainty is not.

The two areas of knowledge therefore relate to certainty in opposite ways. Mathematics can achieve certainty, but it is certainty about what follows from chosen starting points, bounded by Gödel's results and by the practical limits of checking long proofs. The natural sciences cannot achieve certainty about the world, but they produce knowledge reliable enough to act on and can state how reliable it is. For a knower, a claim of certainty should always prompt a further question: certain relative to what? A theorem is certain relative to its axioms. A scientific result is well supported relative to its evidence and its assumptions about error. The two areas also meet. The mathematics behind satellite navigation is certain as mathematics, but whether it describes the clocks on the satellites is a scientific question, and the system works only because corrections for relativistic effects, established by measurement rather than proof, are applied to those clocks.

To a large extent, then, certainty is attainable in mathematics, though it is conditional and internal to the system. In the natural sciences it is attainable to a small extent at most, and pursuing it would damage what makes science reliable: its readiness to change its mind. This is not a weakness of science. It reflects the difference between knowing what follows from what and knowing what the world is like.

References CERN (2012) CERN experiments observe particle consistent with long-sought Higgs boson. Press release, 4 July. Gödel, K. (1931) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198. Hales, T. et al. (2017) A formal proof of the Kepler conjecture. Forum of Mathematics, Pi, 5, e2.