# Is Mathematics always correct?

It seems Mathematical theories/Laws/Formulas are the least questioned in all of the sciences. Is mathematics that good at being closest to the laws of universe, or is it just a logical tool of our own perception of the universe (that being the reason it always works)?

I'll elaborate second part of question above, I thought after reading some answers:

Our natural sense of telling one from many, larger from smaller, numerous from scarce, bright from dark, close from distant, familiar from strange, similar from different and so on, does not needs mathematical axioms and derivation. Even Animals are known to be capable of this. This most primitive logic, the seed of perception is hardwired.So, If Mathematics, itself is based on this sense, starts from here forward, Can we throw "observable phenomenon" and "Physically verifiable" out of the window, when talking about Mathematics ? Isn't this unquestioned nature of our hardwired sense, makes its logical treatment undisputed too ?

Finally,

Perhaps this matter is already in debate.There's a strange, incredible certainty to Mathematics, not found in other sciences. It's still a mystery to me.Though I still think that all sciences are parts of the same universal thread , we got hold of, at different points.

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• Subject of much debate over the years/decades/centuries.... Unlikely to be settled here. – Gerry Myerson Sep 21 '12 at 5:54
• The following quote by Einstein is apropos: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." – Arthur Fischer Sep 21 '12 at 6:57
• I've moved this from math to philosophy, because although it got a lot of good attention from math, it was closed as off topic. Perhaps it will get a different perspective here.\ – davidlowryduda Sep 21 '12 at 15:06
• @fischer: great quote from einstein. – Mozibur Ullah Sep 22 '12 at 12:33
• @wingman: "Mathematics is not about being correct or wrong, it is about being consistent." A statement inside a theory is correct or incorrect or unknown relative to other statements in that theory. If it is proven to be consistent with other statements, then it is correct, if it is proven to be inconsistent, it is wrong, if it is undecided, it is unknown. You can then apply such theories to real life to do probably describe some phenomena, in which case then it is appropriate model for the phenomena, else you try to find some other models, and probably invent a new theory for that. – user2503 Oct 5 '12 at 9:00

## 15 Answers

Physical sciences rely upon thinking of hypotheses and testing them with experiments. The conclusions from physical sciences are always scrutinized because it is the way of the scientific method. In order for a scientific theory to become better, first a deficiency in the theory is discovered, followed by an altered hypotheses, followed by re-testing.

Some people unfortunately see this method as evidence that science is often wrong and unreliable. Science is however a methodology that involves constant refinements of hypotheses to get a clearer and clearer picture of the truth. Therefore science isn't wrong, but the hypotheses science produces are never 100% right either. It is the nature of the game.

Mathematics however is a completely different game. Mathematics works from axioms upwards. Therefore mathematics doesn't have to constantly refine itself as science does. Mathematics is based on foundations, known as axioms, from which the rest of the subject is built from. Unlike in science, the axioms of mathematics are unchanging.

Science can be seen as working in the opposite direction as mathematics. That is, determining the principles from the results, which is much harder than determining the results from the principles (mathematics).

• No, the beauty of mathematics is axioms are assumptions. So all mathematical conclusions are of the form, assuming A then B. So the conditional nature of maths means it can never be wrong. The axioms can be wrong and don't have to match the universe for the conditional mathematical conclusion to be true. – Kenshin Sep 21 '12 at 6:18
• No, I tried to emphasize that axioms are not "the most primitive form of our perception of things around us". I can develop a mathematical theory on false axioms, and the mathematical theory would still be true, because of its conditional nature. Mathematics doesn't depend upon the real world around us, that was my main point. – Kenshin Sep 21 '12 at 6:35
• The difference between mathematics and physics, is in maths, we are free to choose the axioms to be whatever we want. In physics, we must deduce the "axioms" or "principles" from what we observe. Since what we observe is always changing as we do more experiments, our axioms in physics are always changing. This change isn't necessary in maths since axioms don't have to match the real world. – Kenshin Sep 21 '12 at 6:37
• "Mathematics works from axioms upwards." A great many people have done a great deal of Mathematics, pure and applied, without any reference whatsoever to any axioms. "The axioms of Mathematics are unchanging." History does not support this statement. – Gerry Myerson Sep 21 '12 at 7:06
• By "axioms of mathematics are unchanging" I meant for a particular mathematical theory. A mathematical theory can be built on whatever axioms it likes, and that is ok. The axioms for that theory don't have to change because of experiments. Of course, new theories built on new axioms are being developed all the time. – Kenshin Sep 21 '12 at 7:09

The only thing you have to assume to be unconditionally true in Mathematics is some minimal logic (and yes, that's despite having axiomatic systems for logic; you still have to use some form of logic to actually define those axiomatic systems). But logic is assumed to be true in any science (because without it, you cannot draw any conclusions).

But apart from logic, all statements in mathematics are ultimately conditional statements on the chosen axioms. For example, take the statement "there are infinitely many prime numbers." How can we know this to be really true? Well, we have a definition of the natural numbers through a set of axioms, and we have a definition of what it means to be a prime number. From those axioms we can logically derive that there are infinitely many primes. But that statement is implicitly conditioned on the axioms: We have to assume that what we are looking at really fulfils the Peano axioms. If we look at something which doesn't, the claim doesn't hold. However, mathematics doesn't look at a specific system. The statement it derives is not "for this real world object we have infinitely many primes." It says "Whenever we have something which fulfils those axioms, we know that we will find infinitely many primes." It also tells you that if we make certain other assumptions (such as that the axioms of set theory hold), we can derive that we'll find something fulfilling those axioms.

This is also why mathematics is so useful in natural sciences: It does not tell us what assumptions are true. But it tells us what follows if certain assumptions are true (and also, if certain assumptions cannot hold together). So if we have for example a physical phenomenon, we can formulate the hypothesis that it has certain properties. This hypothesis is not part of the real world, but a set of assumptions. Therefore we can now go to mathematics, which tells us what to expect from systems with such assumptions (and also, which additional assumptions we might want to make). Note that this step is completely independent of reality. After we've found what to expect if those assumptions are true, then we can go back to the lab and check if our experiments show the behaviour we just have derived from our assumptions. If yes, we've got a confirmation and may be more confident in our hypothesis, otherwise we have falsified our hypothesis and have to modify it (and again, mathematics will tell us what assumptions will be compatible with our new knowledge from the experiment).

Note that there's another type of questioning theories which is done in mathematics as well as in natural sciences: Namely the questioning whether your results are actually correct. In mathematics, this means checking that there are no errors in the proof (and in some sense this is similar to the experimental tests of theories in natural sciences: We are confident in a proof if it has been sufficiently looked at and nobody has found an error), in physics it means checking that there's no error in the measurement procedure (that is, we really have measured what we thought we measured) and no error in the application of mathematics (that is, we correctly applied the tools we got from mathematics and made no hidden assumptions, and thus our conclusions about what to expect are correct).

• Strictly, you don't really need "logic" per se if all you want to do is arithmetic and mechanical computation. You can, for example, build the untyped lambda calculus out of extremely primitive symbol manipulation (which is not particularly logical and does not need any notion of truth). You can then use Church numerals to do basic arithmetic, or in principle any computation (including logic, natch). – Kevin Jul 31 '18 at 17:40
• @Kevin: How do you define symbol manipulation without using any type of logic? Note that logic ≠ formal logic. – celtschk Jul 31 '18 at 19:47
• Good question. How does logic do it? Just do that. – Kevin Jul 31 '18 at 21:07
• @Kevin: We are already born with the ability to use logic. It is what enables us to think. As far as I know, we are not born with the ability of symbol manipulation. – celtschk Aug 1 '18 at 12:48
• We are not born with formal logic, which indeed requires a basis in symbol manipulation. If you can build formal logic out of symbols, then you can do the same with the lambda calculus. If you cannot build formal logic, then you cannot do math. – Kevin Aug 1 '18 at 15:25

Mathematics is often taken as a kind of path to true. But its methods are not as simple as is popularly made out.

Although mathematical systems are often described axiomatically, this is not how these systems are born. Its often their final form, or rather the form that they are expressed in to bring out their most important properties and to make it look as though they are almost inevitable. Though this is as much psychological for a certain kind of mind.

An example is calculus: Archimedes investigated integration synthetically but could not put it on a formal axiomatic system ala euclid. Its development stalled until Newton/Leibniz utilised the coordinitisation of geometry to begin to fully realise its capability. It was of course noticed that these 'fluzions' were not fully rigorous, and Berkelys criticism of 'ghosts of departed quantities' stung. It wasn't until Cauchy developed the idea of a limit that the foundations of calculus began to be put on a rigourous basis. Now there are a plethora of different axiomatics for the calculus: Synthetic Differential Geometry, Nonstandard Analysis, Diffeological Spaces. Which one of these is the one true & correct axiomatic framework?

Similarly with the more well-known story for Euclidean geometry. The fabric of space-time is much better modelled by Lorentzian geometry.

One could argue, that the axioms are derived empirically, by understanding what important questions can be cast into this kind of language, but surely logic remains a priori.

Again, this is not so simple. We have classical logic from the time of Aristotle which affirmed the law of the excluded middle, (but he noted that this didn't hold for future events), this was eventually formalised as boolean logic, but Brouwer advocated intuitionistic logic that doesn't (his supervisor advised him to establish his reputation in some traditional area before advocating such startling views). People are now researching logics where the law of non-contradiction doesn't hold, where time and modality is taken into account, and so on.

The nature of mathematical truth is not simple. Nor has it shown to be always true. There is a great deal of truth to what social constructivists maintain, that mathematical truth is socially constructed, but that doesn't mean to say that it is solely that, and that it doe not have some sophisticated relation to reality too.

This is what Felix Klein had to say (he was a mathematician famous for formulating the Erlangen program amongst others):

Quite often you may hear non-mathematicians, especially philosophers, say that mathematics need only draw conclusions from clearly given premisses and that it is irrelevant whether those premisses are true or false – provided they don’t contradict themselves. Anybody who works productively in mathematics, however, will talk in a completely different manner. In fact, those people base their judgements on the crystallized form in which mathematical theories are presented once they’ve been worked out. The research scientist, like any other scientist, does not work in a strictly deductive way but essentially makes use of his imagination and moves forward inductively with the help of heuristic aids.

• I have Klein's "Development of Mathematics in the Nineteenth Century" on my bookshelve. Going to read it. You could find my answer below interesting I guess. – Riga Feb 10 '13 at 21:59

There are contradicting assumptions in mathematics, which cannot be resolved, and it is fine! Euclidean and hyperbolic geometry are based on different sets of axioms, which cannot be true simultaneously. However, both geometries are meaningful and have real-world applications.

Now, mathematicians also deals with definitions, and there are of course different ways to define the same thing. Now, it took a while to actually define things like limits, groups etc. and they have looked slightly different throughout history. Some things become considerably nicer with a "better" definition. Some prefer to use $2\pi=\tau$ as THE circle constant which everything is based on, and a lot of formulas becomes simpler using $\tau$ instead of $2\pi.$

The Socratic method: ask the questioner what he means with his words:

• What do you mean with "laws"? There are thorough (mathematical) definitions of what a "theory" is and what a "formula" is, but what is a "law"? Can you tell the difference between a "law", an "axiom" and - say - a "definition"?

• What do you mean with "questioned"? How and why are other-than-mathematical "theories/laws/formulas" more questioned than mathematical ones.

If you happen to give at least partial answers to these questions, it seems worthwhile to continue the talk.

• In fact, "What is Counting ?" can spawn a study of its own. – user2411 Sep 22 '12 at 6:41

I think the answer to this question lies in the distinction: Science deals with observable phenomena where mathematics deals with abstract notions such as numbers, sets or the nature of computability.

Where science strives to be able to express the true state of the universe, mathematics strives to create consistent systems of thinking. When one speaks of a scientific theory they mean a developed and tested explanation of the natural world which can produce falsifiable predictions. When one speaks about a mathematical theory they mean the current state of exploration into one of these abstract notions. A scientist advances his field by testing hypotheses. A mathematician advances his field by proving theorems.

Mathematics does not claim to be the law of the universe, mathematics doesn't claim to be any one thing at all. It happens that science uses mathematics with the hope that the universe is a system that can be expressed with consistency because if not, how would we?

Mathematics certainly can be wrong in that a mathematician presents a faulty theorem with an error in its proof, and it passes the scrutiny of peers and is commonly accepted as true.

Of course after a time the error will be found and the necessary corrections made. Any theorem that follows the rules up from the axiom is correct. It may be totally unrelated to physics or workings of our universe, or it may be related and very similar but with important deficiencies, still, within its own framework it's correct as long as no (stupid) mistakes have been made along the way.

Now an interesting point is some branches of mathematics use theorems without proofs. Famous mathematicians offer a hypothesis with a faulty proof, with known fault - the proof covers a large part of cases but some remain unproven. Now the mathematics follows by building upon that theorem, always with a little disclaimer "Assuming X's theorem is correct", and meanwhile there's a race between enthusiasts to produce a full proof, or alternatively disprove the dubious theorem. In these cases mathematics can be wrong, but only within range of the disclaimer.

The most credible answer I know is given by Henri Poincaré in his "Science and Hypothesis":

He writes about reasoning by recurrence as an example of a true scientific value that is different from tautology. Then he compares mathematics with physics in this aspect:

It cannot escape our notice that here is a striking analogy with the usual processes of induction. But an essential difference exists. Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction — i.e., proof by recurrence — is, on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself.

http://www.brocku.ca/MeadProject/Poincare/Poincare_1905_02.html

Mathematics can only answer limited questions. All mathematics uses deterministic equations, there is no non-deterministic mathematics. We can only solve for 1 variable while holding other variables constant. This is not how the real world operates. The classical three body problem in physics is an example of this. Other examples are flow dynamics and chaos.

• How can you say there is no non-deterministic maths and then give examples? Probability theory is full of non-deterministic equations. Is the mathematics of quantum theory not maths? It looks like supersymmetry and octonions are deeply linked, and and it all emerges naturally from the mathematics of uncertainty – CriglCragl Aug 2 '18 at 21:05
• @CriglCragl uncertainty is not the same as non-deterministic equations. You are conflating different ideas. How many variables are you solving at the same time in one equation?? – Swami Vishwananda Aug 3 '18 at 4:43
• Evolution and consciousness are en.m.wikipedia.org/wiki/Nondeterministic_algorithm being solved for multiple outcomes simultaneously. You think that is beyond maths? – CriglCragl Aug 3 '18 at 9:59
• @CriglCragl to quote the wiki article - "The nondeterministic algorithms are often used to find an approximation to a solution, when the exact solution would be too costly to obtain using a deterministic one." - these are computational algorithms, not mathematical formulas. they can only find approximations, and the article states they are not 100% correct. – Swami Vishwananda Aug 5 '18 at 8:04

No, mathematics is not always correct. There have been plenty of false theorems and proofs. To mention only few:

In 1833, the year of his dead, Adrien Marie Legendre presented an overwiev of proofs of the parallel axiom to the French Académie des Sciences. It included six rigorous proofs, three of which using infinite angular areas. Here "rigorous" is to be understood in the meaning of his times as present mathematicians use "rigorous" in the meaning of our times. But obviously there can never be absolute rigour, neither then nor today.

The Schröder-Bernstein theorem was repeatedly stated (and claimed as proven) between 1882 [G. Cantor, letter to R. Dedekind (5 Nov 1882)] and 1895 [Cantor's collected works, p. 285] but has never been really proved by Cantor. This theorem is called after Ernst Schröder and Felix Bernstein, because both proved it. Alwin Korselt however discovered a flaw in Schröder's proof in 1902. Alas the Mathematische Annalen did not publish the correction before 1911. [A. Korselt: "Über einen Beweis des Äquivalenzsatzes", Math. Ann. 70 (1911) 294] Nevertheless it took some time until this correction received public attention. Ernst Zermelo noted in his edition of Cantor's collected works as late as in 1932: "The theorem [...] has been proved only in 1896 by E. Schröder and 1897 by F. Bernstein. Since then this 'equivalence-theorem' is considered of the highest importance in set theory." [Cantor's collected works, p. 209] We learn from this that wrong proofs can survive in mathematics over many decades.

Present set theory is considered the fundament of mathematics. As Fraenkel put it: "If the attack on the infinite (the finished infinite of set theory) will succeed ... only remnants of mathematics will remain." It can be shown however that set theory is in conflict with mathematics. Simplest example: McDuck who daily receives 10 \$ and spends 1 dollar will become infinitely rich according to analysis but will go bankrupt according to set therory.

There are many more proofs that mathematics is not reliable. But these few should be sufficient.

Of course you can say that mathematics is only the pure nucleus stripped off the human errors and mistakes. But how would you ever know then what this nucleus of mathematics is, in particular with respect to Gödel's results?

In my opinion (I am a 10th grader in Turkey yet I am also a math nerd), if you are looking through the eyes of a mathematician and see a correct result derivable from a set of axioms that we have accepted that does not suggest a paradox. (Yes I am aware of Gödel's incompleteness theorem.) That suggests that mathematics is the closest thing we have to perfection.

The rigorousness of mathematicians is unparalleled in the scientific community. Mathematicians always require proofs to any and all conjectures. Some important questions like the Goldbach conjecture and the Riemann hypothesis have trillions of examples and no counterexamples and yet mathematicians don't accept them as facts but questions nonetheless. In any other science they would be seen as facts, yet mathematicians don't see them as facts. (That is one of the reasons that I want to be a mathematician, not a doctor or a biologist or even a businessman.)

Yet when you consider the real world, things get messy. Even if every theorem that you have utilized and every calculation that you have made is true, your results may not be true because the model that you have used to describe the world was incorrect, and trust me modelling the world is quite hard.

For example, some of Newton's laws are wrong. (They are not perfect, to be precise, yet really, really good for everyday use.) They are wrong when we are looking at objects small enough or going fast enough. Yet we make space shuttles and fighter jets using them not because they are perfect but because they are a good enough approximation.

Yet if you are using those laws to build a GPS without considering relativity, you will fail. While some of the best GPS systems measure the margin of error in millimetres, without compensating for relativity you would have kilometres of margin of error.

I may be getting a bit off topic here but I will say it nonetheless. Consider that science wants to quantify and make things as repeatable as possible. Making things quantifiable and repeatable are perfect ways to describe mathematics. No matter how you feel today nor how close you are to the event horizon of a black hole, if you plug in an x-value to an equation you will get the same results, which is what scientists need when modelling the world.

• I made some edits. You are welcome to roll them back or continue editing. You can see the versions by clicking on the "edited" link. I see you have quotes to questions in the Physics SE in a comment. You might want to edit the post and insert them into your answer to support it. The more you can support your opinions the better when giving an answer. – Frank Hubeny Aug 1 '18 at 0:50

Mathematics is completely wrong; and to prove that we must define wrong or false first. Truth must be defined in the following way – (a) Laws of nature are the only truths (b) these laws are created by the objects of nature and by their characteristics (c) Nature always demonstrates its truth.

Consider a simple mathematical statement (M1) 1+2=3. Everybody will understand M1, one orange and two apples give us three fruits. But it is completely wrong use of mathematics for several reasons. The numbers 1, 2, 3 are points on the real line; they cannot be apples and oranges. Points are not objects of nature. Therefore real numbers are false. Also these points are defined as points on a straight line, called real line. But there is no straight line in nature, because all objects of the universe are continuously moving. Therefore fundamental definition, straight line, points, etc., are all false and do not exist in nature. Therefore such mathematics can never work for nature and engineering. There are many examples to prove that math cannot work in nature. Take a look at chapter one on truth in the free book on soul theory at https://theoryofsouls.wordpress.com/

• Do you have references to other philosophers who take a similar view to yours? This will help support your answer and make it less of a personal opinion. – Frank Hubeny Jul 31 '18 at 15:29
• Yes, of course. The soul theory book I have mentioned provides the details. The book has about 180 references. Moreover, any mathematics book on Real Analysis will explain the same thing. The things that I have mentioned are also quite obvious. For example – how can there be a straight line when everything is moving? Take any two points in space, join an imaginary straight line to them, then take a middle point on the line. You will see, within moments, the midpoint will no longer be on the line, because everything, including the midpoint, is moving. – Subhendu Das Aug 1 '18 at 18:46
• I know you have the references in the book, but 180 are also too many references. One needs only a relevant handful right in this answer to strengthen the answer. You are saying that mathematics is "completely wrong". This is not what people expect. Who else describes mathematics that way--as completely wrong? – Frank Hubeny Aug 2 '18 at 13:23

Theorems can always be derived from axioms that we assume are correct. There are no "correct" axioms. You can select anything that you want, yet they must not contradict themselves or other axioms. If you have a particularly good set of axioms you may not even have contradictions, but that is impossible to prove. Therefore mathematics is the most reliable tool humans have ever produced. (Yes even more reliable than an AK-47 or an HK MK23.)

What mathematicians do is create an idealized world where the only forces affecting the ball that you have thrown are the force that you have applied and the force of gravity. It always follows an idealized path. Yes, you may lose some accuracy, but it is good enough for all practical purposes. If you need more accuracy you can also consider air resistance, the movement of the earth, etc.

• I made an edit which you may further edit. – Frank Hubeny Aug 1 '18 at 1:11

From an Intuitionist point of view, mathematics is a science, and it evolves like any other science. But as a science, what mathematics studies is not what we naively understand to be its proper domain.

The objects of mathematics that mathematicians prove things about are not the scientific object of the discipline, they are its experiments and its technology. The subject matter is, instead, the intuitions of human beings. Mathematics determines how those intuitions fit together or contend against one another, and in what ways our naive natural assumptions about how they will combine are borne out. We test those things in the experimental process of writing proofs.

All the past experiments of physics remain experiments of physics, and all the technologies that result from the applications of past physics also remain valid, even when the physics on which they originally relied is modified. Likewise, all past proofs and techniques of mathematics remain proofs and techniques of mathematics. What changes and gets refined at the same rate as the laws of physics develop is the selection of what areas of mathematics are interesting or applicable to other sciences.

In that capacity, mathematics is really a branch of psychology. It studies what intuitions are readily evoked in different combinations in a wide range of humans, and are therefore available to use in abstract explanations. We can be wrong about what will make sense to elaborate, or what will have applications to our other mental structures, versus what will take too many forms or will simply be pointless elaboration, even if the mathematics itself is never 'right' or 'wrong', but just 'there'.

As noted in another answer here, it is quite reasonable to look at all of mathematics as fictional, and therefore false, but internally consistent. And it loses none of its value if this is the case. Because it is not, at root, about truth. It is about conceivability: about what can potentially make sense to a human mind, and what ideas only seem to be usable, but when pressed will ultimately not hold together.

• Ether? Caloric? And what about the theorem Fermat is thought to have had in mind, that proved not to work? 'Part of' only as historical footnotes – CriglCragl Aug 2 '18 at 21:10
• @CriglCragl And that means what? Those theories are disproven, but the actual experiments that were done, that led to their formulation remain in the corpus of science and must be explained by any later theory, if only as misinterpretations. Yes, some proofs, like other forms of experiment are actually flawed. And they can be dismissed when the flaws are pointed out. So what? You seem to have some objection but I cannot address it unless you actually articulate it. – jobermark Aug 2 '18 at 21:22
• Statistical mechanics needed to explain all of Carnot's results. Quantum theory needed to explain or reinterpret all the behaviors of light as a wave that cause the theory of Ether to be proposed. Experiments remain facts of science. – jobermark Aug 2 '18 at 21:26
• Wrong ideas, wrong experiments, wrong observations. They are mysteriously in 'the corpus'. What about what is forgotten? That too? Seems unfalsifiable, and beyond Occam's razor. Science is a process, not a ledger – CriglCragl Aug 3 '18 at 10:03
• Sorry, physics, not maths – CriglCragl Aug 3 '18 at 10:50

There's a lot of wide-eyed and giddy enthusiasm for maths here. It would be more appropriate to describe mathematics as formally systemised thoughts, as a language. Godel and the failure of the Hilbert program showed that mathematics isn't a ladder to a god's eye view, butva floating point, defining up and down - and as Hofstader described looping: https://absoluteirony.wordpress.com/2014/09/17/nagarjuna-nietzsche-rorty-and-their-strange-looping-trick/

Mathematics can never get round https://en.m.wikipedia.org/wiki/Münchhausen_trilemma Where do axioms come from, and how do you know they are right? Only by the interesting behaviour of the resultant system. Mathematics' dirty little secret.