Does Popper's theory of falsification apply to mathematics?

Question

Mathematics is generally & popularly judged a science in the basic duality: science - humanities. As enemies and collaborationists. The border heavily & fiercely policed.

However, it seems to me that Poppers theory, which entitles science-hood by falsification doesn't apply to mathematics at all.

What can it mean that the Number Theory is falsifiable? Certainly a tightly-focused question will either be true or false. More general conjectures & ideas will be true when enunciated as the mathematical landscape is seen and a new shape formed. For example the Langlands programme (higher dimensional representation theory). Significance through aesthetics & ethics seem the key theme. The serious intent (ethic) towards the good & beautiful (aesthetics) towards the reverance & delight of contemplation. Platonism in essence.

Badiou characterises knowledge as four domains (conditions) - love, science, art & politics.

Is it then love - Number Theory being the material incarnation of a mathematicians embrace and adoration of Number?

If not then is it art - Number Theory being the glorification of Number through steady & inspired craft. As a cathedral to God, so Number Theory to the One?

If neither then could it be politics - creating harmony amongst bickering wilful abstract entities intent on having it their way? Number theory being a nation of number systems.

If none of these, then does it lie with Philosophy, the place from which these four conditions converge (in Badious system)? One is reminded that Badiou states that mathematics is the very ontology of philosophy. He may give it space there, but does anyone else?

Answer 1

It is the other way around. Falsification is just a statistical form of proof by contradiction. In proof by contradiction, you start off by assuming that a premise p is true. Then you show that such an assumption leads to a conclusion q, which has already been shown to be false. So long as we assume that our axiomatic system is logically consistent, this result implies that p is false: p -> q and ~q together imply ~p.

In falsification/hypothesis testing, we start with a theory T, and then we make an observation O. From T we can determine how likely O is. If the probability of O given T is low, then we can say that the probability of T given O is low. Of course, if the probability of O given T is 0, then we end up with an absolute falsification which would be exactly the same as proof by contradiction.

Of course, in falsification, we rarely have absolutes, and even worse, we are not really testing a single theory but we are using a whole body of theory to come to our conclusion. This result makes since a bit more messy, and I discuss this issue in more detail here.

Answer 2

After the reconstruction of foundations of mathematics in 20 century, i.e. getting rid of the famous paradoxes (like Russell's paradox), the modern mathematics is based on the belief that no other paradoxes will appear again. Despite numerous efforts, logicians did not manage to prove that the systems of axioms of modern set theories are consistent (and at the same time they did not manage to find new paradoxes). This means that theoretically it is possible that in future somebody will find a new paradox, and this will have the corollary that some mathematical results (possibly most part of them) turn out to be false. This can be considered as the evidence of the falsifiability of mathematics.

Answer 3

Of course it doesn't apply. It only applies to natural science. And even there, it doesn't apply everywhere. In a strict sense it applies only where you can conduct experiments in a controlled environment, i.e., test hypotheses.

This, IMHO, excludes, for example, the theory of evolution. While this theory does a great job of explaining the development of life, it can not make any predictions. Hence, there is nothing to test and so it doesn't belong to Popper's domain.

Answer 4

although often called "The Language of Science", Mathematics is not the same as Science (or, more specifically, Natural Science).

Mathematics is a specific application of philosophy and logic to the concept of quantity. whereas in Science that quantity of "stuff", whatever that stuff is, is salient, the techniques of mathematics is necessary to understand more fully what the quantities are, how they may be related, and to speculate in how some quantities may be related to others. in that, there are predictions of how specific physical or natural quantities will behave, experiments devised, observations made, and these predictions are supported or falsified. scientific theories can be developed and supported from that.

but there is no such counterpart in pure mathematics. sometimes conjectures are made (such as Bertrand's postulate or Fermat's Last Theorem), but they are just conjectures until they are proved by derivation, not by observation as is done in science.

now these two conjectures, now proven mathematical facts will never be disproved in the future. that is also different than falsifiability in science. General Relativity at this time seems like the unrefuted truth, but in another time, so was Newtonian mechanics and gravitation, but now we understand that to be a very close approximation to the truth for slow relative speeds and not in extremely intense gravitational fields. so nothing in Science has the status of never, ever, being falsified.

but in mathematics, a properly proven theorem will never be disproven given the same axioms to the theorem.

it's different. mathematics and natural science are not the same things. mathematics, although a salient tool in science, is not science. Popperian falsifiability applies to science. it is a demarcation of ultimately what is science and what is not. not all scientists agree with that demarcation, but i do.

Answer 5

Absolutely the concept of falsifiability applies to mathematics (at least in a roughly metaphorical way, and maybe much more). I am not a big expert on Popper but ever since hearing it, I think it is possibly more relevant/applicable to math than almost any other science/field in the following sense.

Math works through the advance of hypothesize, test, and verify. In this way it is no different than any other scientific field, and in many ways embodies all the others in a more pure form. The hypothesis is typically called a "conjecture". There are many open conjectures, some quite famous, and very old. (e.g., there are Greek number theory problems over 2 millennia old), but a single counterexample is a falsification of a conjecture. Notice that one does not need to do complex experiments to refute or replicate a falsifiable claim. It's all abstract, cerebral/mental.

However, there is also nowadays even an experimental/empirical falsifiability going on in math, in increasingly widespread/significant use: computers are used to explore conjectures and find "not too big" counterexamples.

Note, however, there are some senses where falsifiability fails in math. undecidable problems do not have algorithmic answers (e.g., Hilbert's 10th problem, undecidability of finding solutions to diophantine polynomials). (In fact it's striking how many parallels there are in the theory of falsifiability to undecidability.)

Some theorems are unprovable. Some problems are extremely difficult and may never be resolved, (e.g., the Riemann conjecture open 1.5 centuries even after intense focus), etc. Also, the issue of replicability of falsifiability rears its head with computer assisted proofs, e.g., the famous 4 color theorem. Can they be trusted?

It is tempting to propose that all well-formed math conjectures are falsifiable and Hilbert had such a belief early on and wrote about it and the idea motivated some of his own conjectures, but of course Godel strikingly proved otherwise.

Another issue that arises is "proofs that were later found to be incorrect" (Mathoverflow). So human fallibility and falsifiability are interrelated. Anyway one might say that mathematicians are more rigorous about and dedicated to falsifiability than any other "scientists".

Answer 6

Mathematics is a branch of metaphysics according to Popper, see "Realism and the Aim of Science" Part I Chapter III. Popper does not judge metaphysics adversely compared to science. Some people want to draw that implication because they uncritically admire everything labelled science as a substitute for making their own judgements.

For a couple of other perspectives on maths that seem broadly consistent with Popper's epistemology, see "The Fabric of Reality" by David Deutsch, Chapter 10 and "Proofs and Refutations" by Lakatos. As a side note, I do not recommend any other material written by Lakatos, who wrote a lot of material attributing positions to Popper that he never advocated.

Answer 7

As a corollary to some of the other answers here, it is worth noting that whether you consider mathematics to be falsifiable is likely to depend on your account of what mathematics is.

On a formalist account, mathematicians are just manipulating strings of symbols according to rules. As such, the mathematics itself cannot be falsified, although there can always be mistakes that need to be rectified. On this view, it is only when the strings are given an interpretation that a falsifiable proposition emerges.

On an empiricist account, the question of whether mathematics is falsifiable goes hand in glove with the question of whether logic is empirical. This is a question on which a lot has been written. Famously, Quine held that all of our knowledge, including logic, is holistic in nature and falsifiable only as a whole, not on a sentence by sentence basis. Consequently, even logical or mathematical statements are revisable in the light of future knowledge - they have no special a priori status - they are just relatively well protected from being revised because they serve an important and central role in organising our web of belief. Also, Putnam wrote an influential paper called "Is Logic Empirical?" in which he proposed that we might want to revise our logic in the light of quantum mechanics.

On a logicist view, mathematics is reducible to logic, so a mathematical statement is falsifiable only in the sense that it might be possible to show that it entails a logical inconsistency.

To an intuitionist, a mathematical proposition is something constructed from the mathematician's intuition. One might therefore reasonably suppose that mathematics is falsifable to just the same degree that a mathematician's intuition is falsifiable.

As to mathematical platonism: I'm not sure that I really understand it, but I suppose that platonists would regard mathematics as falsifiable, just not empirically.

Answer 8

I will continue to argue my tiny minority position that mathematics is a branch of psychology.

Mathematics is still science. It is the science that tries to identify the thought processes and forms of idealization that humans share, that improve our leverage upon thought and science.

Mathematical disciplines do not suffer from conflict with the facts, but they clearly descend into irrelevance when the facts they apply to do not matter. What is false, when math is 'falsified' and becomes irrelevant, is the idea that a given question is interesting to the world at large, or has use because it captures an important insight.

Answer 9

Since Theorems in Mathematics are those formulas that are proved and every proved formula is true (Completeness of 1st order logic), the falsifiability can not be applied to Theorems in Mathematics.