## Was mathematics invented or discovered?

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118

What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered?

Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity?

2Just to make it clear: discovery = finding something that existed before (e.g., a frog, black holes) invention = intellectual creation (e.g., of a system) – fubra – 2012-09-25T09:33:30.127

3what if invented = discovered? – Ooker – 2017-04-16T16:33:15.877

My first reaction was "man, don't ask me a question like that", so great question...lol! But I would say it's BOTH! Certainly, there was imagination involved in the discovery and the unfolding of those processes relates to the spontaneity of genius in the establishment of reaching great mathematical solutions. – Paradox Lost – 2012-08-15T21:42:50.963

It's a valid historical question. Philosophically the question poses a false dichotomy. It is likely that we started a system of counting in our head that gradually evolved into the abstract system of symbol manipulation that we have now, so it was probably a continual interaction between the two until the formal system emerged. – TheDoctor – 2017-07-13T18:44:55.837

@Ooker, indeed; no one discusses that possibility. There is indeed a philosophy which poses that, and it even has axioms (though not in the sense that modern mathematicians uses them—more like Euclid's definition). – Wildcard – 2017-08-29T00:36:22.380

What we invent is called application. And what we find already existing is called discovery. Mathematics is the logic of understanding the facts through measures. Here measure is only a medium of understanding. So, I feel mathematics a discovery..........!!! – Aneosh – 2012-11-19T19:10:45.630

@Ooker I wouldn't say so. As discovering means finding out something that exists naturally, like gravity. You cannot invent gravity, at most you can invent a machine which can mock it. They are really different. Since maths existed even before humans started to make calculations, it means that it is discovered. The electrons didn't start to spin around protons and neutrons after we found out that they do, in an orbit(which has a ellipsis-like form) – Red fx – 2018-02-02T11:06:05.323

@Redfx are dinosaurs invented or discovered? Surely we discovered the fossil, but for an imaginary human leaves at the birth of the Earth, they are invented, right? – Ooker – 2018-02-03T01:20:41.557

@Ooker as you said yourself, it is neither invention nor discovery, it is just imagination. It has nothing to do with these concepts. But lets say they really really believed in this and then they figured out that the statement was really wrong, then it would be the discovery of the truth. Because you perceive 'something' that was there or not there. In very simple words, you discover something that is already there. But you invent something unusual. Like you cannot invent 'dark matter', at most you can discover it is there or it is not there. – Red fx – 2018-02-07T08:48:09.620

@Ooker dinosaurs are so discovered in the sense: we discovered that there were these kind of living creatures before us. We didn't create them, or we didn't put them in the past ourselves. [my prev. comment on imagination was about the human birth with leaves if you meant that] – Red fx – 2018-02-07T08:54:58.437

As of December 21, 2018, The World Has A New Largest-Known Prime Number: There's a new behemoth in the ongoing search for ever-larger prime numbers — and it's nearly 25 million digits long. https://www.npr.org/2018/12/21/679207604/the-world-has-a-new-largest-known-prime-number?utm_campaign=storyshare&utm_source=facebook.com&utm_medium=social&fbclid=IwAR3JPtG_toEtR1XiZsZU_OvEOkD2vH17uGZmDTPM1PughJLSMsVRnZr4uec

Were Bachelors (the people) discovered or invented? It's kind of a moot question. – None – 2019-06-18T17:26:34.767

Math is invented through consensus on naming, axioms, logic, etc. The implications within these systems can appear to be discoveries but they're also an inherent result of the invention. – sfmiller940 – 2019-07-11T18:02:03.643

32

Here is a headline from 2008: http://news.bbc.co.uk/2/hi/americas/7640183.stm Huge new prime number discovered I wonder who would say that this prime number was invented in 2008?

– GEdgar – 2011-06-26T02:41:28.897

8Mathematics is a language. Just as we didn't invent a tree, we can describe a tree using English, or French, or Mathematics. – ProfessorFluffy – 2015-04-17T13:57:42.470

9@GEdgar Intuitionists do not claim that individual numbers are invented - in fact an infinite number series is a principle of intuitionism. They would say that it became true that the number is question was prime. Prior to that, it was neither true nor false that it was prime. – adrianos – 2012-01-06T13:54:44.633

1@ProfessorFluffy But did we invent English, or discover it? – None – 2015-11-06T16:58:57.363

@nocomprende Well to the point -- We converged upon it. If Mathematics is a structure made of shared intuitions, they become shared by some process. I would blame evolution and a little bit of culture. In that case, the answer is neither: it was gifted to us by an outside force. – None – 2016-03-22T13:55:51.860

@jobermark I would say the answer is neither also: no two people can ever know the same thing, there are no objects or truths and language, time and causality are delusions. Mathematics goes "poof!" Except for all the people riding in elevators. – None – 2016-03-22T23:11:36.830

If maths were invented, it wouldn't be possible to draw a circle without knowing the value of Pi ... but it is. – user3646932 – 2016-12-29T21:38:36.427

139

"Intuitionists" believe that mathematics is just a creation of the human mind. In that sense you can argue that mathematics is invented by humans. Any mathematical object exists only in our mind and doesn't as such have an existence.

"Platonists", on the other hand, argue that any mathematical object exists and we can only "see" them through our mind. Hence in some sense Platonists would vote that mathematics was discovered.

@eMansipater: I am not an expert on this topic. All I can refer to is probably wiki which I have now added. – None – 2011-06-07T20:20:35.507

1Interestingly and somewhat ironically, pragmatism would say this is the only important answer. – None – 2018-05-09T13:27:04.293

9True Platonists would argue that anything we learn is in fact remembered. This is the point of Socrates walking Meno's slave through a simple Euclidean proof about squares -- i.e., that through dialogue and introspection, our 'innate' knowledge (memory!) of mathematical Reality can be recovered in some partial way. – Joseph Weissman – 2011-06-07T22:42:52.793

@Chuck: suppose somebody, whose vision is limited to 2-D objects, asks you: "Are Egyptian pyramids triangles or squares?" Would it be acceptable for you to answer: "Well, according to Plato, who lives in a spaceship and looks down on Earth, they are squares; according to Aristotle, who stands on the ground, they are triangles; but IMHO they are a little bit of both." – Michael – 2013-10-02T14:13:40.507

This is hardly an answer. Completely a comment. I wish I had enough reputation to downvote. – something – 2013-11-16T02:42:37.827

4Good enough for my upvote--I just want readers to realise these are specific schools of thought with defined characteristics. – eMansipater – 2011-06-07T20:27:09.407

What do you mean exactly when you say that - for the Platonist - we can only see mathematical objects through our mind? Do you have in mind some kind of Godelian faculty of intuition? If so, I fear this isn't really the acknowledged understanding of the Platonist's position. If not, can you make it clearer? – Mathmo – 2014-03-29T01:06:03.793

3I think it's fair to say that the number of professional mathematicians who apply instuitionist logic are in a small minority, and that the vast majority have no qualms about, for example, using the law of excluded middle and non-constructive existence proofs. – AndrewC – 2014-08-08T07:16:47.107

12Ok, good definitions; but what is your actual answer? – Geoffroy CALA – 2011-10-11T23:03:43.710

This is one of my favorite philosophical debates. – Jimmy360 – 2015-03-09T13:09:46.500

53I don't think this is a good answer. It reminds me too much of academic philosophers' evasive ism-dodges, i.e. when you ask them 'Does you argument fail because of X?' and they respond: 'Well if you're a Y-ist then no and if you're a Z-ist then maybe yes, but I'm not sure which one I am yet etc etc.' - The schools themselves mean nothing and people who haven't explored the issue independently should not, I believe, get told to look up a certain school of philosophy as their point of first contact – Chuck – 2011-06-09T21:00:21.150

1Mathematical platonism isn't intended to be the same as Plato's views but just similar in certain respects. That's why its often spelt with a lowercase P. – adrianos – 2012-01-06T13:24:59.300

2@GeoffroyCALA Why does it matter whether any one individual is an intuitionist or a platonist? – adrianos – 2012-01-06T13:25:54.383

2@Joe: That's quite correct. Plato (via Socrates) held that we knew everything from previous 'afterlives' and that we remembered them, but thought we were learning something new. The passage with the slave boy sets this out very clearly. – boehj – 2011-06-13T03:48:20.220

@boehj To which passage do you refer? – p.a. – 2012-08-16T07:14:18.793

86

My personal point of view is that mathematicians invented the axioms and the rules of operation, the rest are discovered. Mathematicians invented the notations for writing down the concepts which are discovered within the universe of an axiom.

The concept of numbers exists, but we invent the notation that the glyph '1' and the sound /wʌn/ refers to the concept of singular object that we discovered. We invented the rules of matrix multiplication, but the consequences of the way we do matrix multiplications are discovered.

Most of the time, we deliberately invent a set of axioms that will lead us to discover a set of facts we want to be true. This is certainly true with imaginary numbers, we invented them so that we can discover the solutions to problems we previously were unable or difficult to solve.

So mathematics is a collection of human invented axiomatic systems, notations and tools (such as the plane) used for exploring the logical consequences and concrete applications thereof? – Alex Nye – 2012-10-09T22:17:02.850

@user21820, every proof I have seen for the uncountability of the reals has a logical error that involves implicitly assuming their existence as infinite uncountable sequences before proving it. This is very distinct from the original meaning of an "axiom" as a truth sufficiently obvious to not require a proof. – Wildcard – 2017-08-29T00:15:12.860

2

@Wildcard: I don't know which proofs you've seen, but the proofs for reals being uncountable usually starts by assuming that real is infinite countable, then proceeds to derive a contradiction based on that assumption.

– Lie Ryan – 2017-08-29T03:34:24.163

@LieRyan, yes, but even Cantor's diagonal argument implicitly assumes that you can have an arbitrary infinite sequence of digits which cannot even be described; with no method or rationale to it whatsoever; and then proceeds to assemble an infinite number of such infinite sequences into another arbitrary infinite sequence with no rationale or means of computation, and then draw a diagonal across it. The unknowability of the entirety of any given real number in the list is already assumed. Try to make the argument using only computable numbers, and a computable list of such, and you'll see. – Wildcard – 2017-08-29T03:44:51.777

In other words, all arguments and all proofs of the uncountability of anything begin from a basic assumption that "there are things which cannot be known" and then proceed to prove that there are things which cannot be known. Philosophically, this is as nonsensical as Kant's "unknowable" and other such musings. You can prove anything which you assume as true. If you do not assume the validity of an infinite set with unknowable elements, you don't wind up with unknowable conclusions. See also http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf on why real numbers are a joke.

– Wildcard – 2017-08-29T03:48:21.480

@Wildcard: No. The diagonal argument can be used to prove that there is no computable list of all computable reals, so your comment is wrong. In the most constructive form, I can write a program that given any input P that enumerates a list of computable reals (encoded as programs), will output a computable real that is not enumerated by P. – user21820 – 2017-08-29T06:02:31.347

As for Wildberger, I will tell you frankly that whatever he says about logic has little value, and only when you truly understand the circularity behind all of mathematics including the purely finitary fragments, will you grasp that mathematics depends crucially on accepting the complete collection of natural numbers. If you want an explicit point where he is talking nonsense, he says "And Axiom 6: There is an infinite set!? How in heavens did this one sneak in here?" which just shows a thorough lack of knowledge of ZFC. I say this despite myself having a strong distaste for ZFC. – user21820 – 2017-08-29T06:09:56.117

@user21820, consider carefully whereof you speak. Why can you not write a program which, given any program which enumerates a list of rational numbers, outputs a rational number not on the list? Again, you assume the existence of completed infinite actions. – Wildcard – 2017-08-29T07:22:08.807

@Wildcard: That is irrelevant. In the most constructive form, I never assumed any completed infinite object. Please read it carefully again. If you do not know how to write such a program, it may be because you do not understand the definition of a computable real. – user21820 – 2017-08-29T07:28:34.603

I've expanded on my point in chat.

– Wildcard – 2017-08-29T20:48:20.910

@AlexNye: sorry for the late response, but yes you've got it right, spot on. – Lie Ryan – 2013-02-27T16:20:52.243

Do you mean that some people don't think of imaginary numbers as really existing but when they see a proof of a statement about real numbers using complex numbers, they can figure out after reading it how to write a proof of it using only real numbers? For example, they might decide that the formal meaning of a statement about complex numbers is really a statement about ordered pairs of real numbers. – Timothy – 2018-12-09T03:20:30.663

Or you do "discover" something or you "want it to be true". They're not always the same. – Rodrigo – 2018-12-22T02:06:42.257

6Complex numbers appear invented, particularly due to the label "imaginary numbers", but both names are slightly misleading, as historically there was great reluctance to accept that the terms involving square roots of negatives represented anything, thus the pejorative name "imaginary", but accepting them was a hugely simplifying asssumption that was eventually come to, and"complex" is unfortunate for something that introduced so much simplicity. It's fair to say that the people who first came across them didn't initially accept their validity, so maybe they're more discovered than invented. – AndrewC – 2014-08-08T07:26:46.663

5With the exception of Euclidean geometry, through most of mathematical history, axioms were laid down long after the corresponding mathematics was developed. Only from the 19th century was the axiomatic method applied to the whole of mathematics. – John Bentin – 2014-08-29T21:47:12.897

32why does the concept of number exist, but the concept of complex number invented? – Artem Kaznatcheev – 2011-09-12T17:42:09.550

4@Artem Kaznatcheev: Good point. Complex number is just a notation for writing down a pair of numbers, we invented the rules of complex arithmetic so they can conveniently represents coordinates in a plane and a few common transformations. The concept of a plane and a point in plane exists, but the notation (e.g. complex number) are invented. – Lie Ryan – 2011-09-12T21:15:26.407

1one could also say that someone found that the particular set of axioms matched the intuition. – Mitch – 2011-10-11T21:05:45.340

We don't invent axioms; the mathematical statements which are the axioms exist independent of us, but we choose the axioms. – user1997744 – 2015-02-23T16:38:02.490

@AlexNye Reducing mathematics to the collection of human invented axiomatic systems, notations and tools is like reducing a language to the collection of human invented grammar rules and letters. Language is that plus a tool to convey meaning. I like to believe that Math is the above plus a tool to explore, analyze and describe mathematical concepts (whatever a math concept actually is). – None – 2016-02-23T22:44:37.213

@ArtemKaznatcheev: The simple answer is that complex numbers were discovered, not invented. Given the natural numbers, you can discover that the rational numbers form a field. Its metric completion (via Cauchy sequences) gives you the real numbers, which you may discover is the unique ordered archimedean field. After that, you may want to have roots of polynomials, and it is an amazing theorem that the complex numbers are an algebraically closed field containing the reals, and in fact the unique algebraically closed algebraic extension. There is nothing to invent here except the notation. – user21820 – 2016-05-24T14:50:34.140

@ArtemKaznatcheev: (Unique up to isomorphism of course.) – user21820 – 2016-05-24T14:51:28.973

80

There are things that are discovered, and things that are invented. The boundary is put at different places by different people. I put myself on the list and I believe that my position is objectively justifiable, and others are not.

### Definitely discovered: finite stuff

By probablistic considerations, I am sure that nobody in the history of the Earth has ever done the following multiplication:

9306781264114085423 x 39204667242145673 = ?

Then if I compute it, am I inventing it's value, or discovering the value? The meaning of the word "invent" and "discover" are a little unclear, but usually one says discover when there are certain properties: does the value have independent unique qualities that we know ahead of time (like being odd)? Is it possible to get two different answers and consider both correct? etc.

In this case, everyone would agree the value is discovered, since we actually can do the computation--- and not a single (sane) person thinks that the answer is made up nonsense, or that it wouldn't be the number of boxes in the rectangle with appropriate sides, etc.

There are many unsolved problems in this finite category, so it isn't trivial:

• Is chess won for white, won for black, or a draw, in perfect play?
• What are the longest possible Piraha sentences with no proper names?
• What is the length of the shortest proof in ZF of the Prime Number Theorem? Approximately?
• What is the list of 50 crossing knots?

You can go on forever, as most interesting mathematical problems are interesting in the finite domain too.

### Discovered: asymptotic computation

Consider now an arbitrary computer program, and whether it halts or does not halt. This is the problem of what are called "Pi-0-1 arithmetic sentences" in first order logic, but I prefer the entirely equivalent formulation in terms of halting computer programs, as logic jargon is less accessible than programming jargon.

Given a definite computer program P written in C (or some other Turing complete language) suitably modified to allow arbitrarily large memory. Does this program return an answer in finite time, or run forever? This includes a hefty chunk of the most famous mathematical conjectures, I list a few:

• The Riemann hypothesis (in suitable formulation)
• The Goldbach conjecture.
• The Odd perfect number conjecture
• Diophantine equations (like Fermat's last theorem)
• consistency of ZF (or any other first order set of axioms)
• Knesser-Poulson conjecture on sphere-rearrangement

You can believe one of the two

• "Does P halt" is absolutely meaningful, so that one can know that it is true or false without knowing which.
• "Does P halt" only becomes meaningful upon the halting of P, or a proof that it doesn't halt in a suitable formal system, so that it is useful to introduce a category of "unknown" for this question, and the "unknown" category might not eventually become empty, as it does in the finite problem case.

Here is where the intuitionists stop. The famous name here is

• L.E.J. Brouwer

The intuitionistic logic is developed to deal with cases where there are questions whose answer is not determined true or false, so that one cannot decide the law of excluded middle. This position leaves open the possibility that some computer programs that don't halt are just too hard to prove halt, and there is no mechanism for doing so.

While intuitionism is useful for situations of imperfect knowledge (like us, always), this is not the place where most mathematicians stop. There is a firm belief that the questions at this level are either true or false, we just don't know which. I agree with this position, but I don't think it is trivial to argue against the intutionist perspective.

### Most believe discovered: Arithmetic heirarchy

There are questions in mathematics which cannot be phrases as the non-halting of a computer program, at least not without modification of the concept of "program". These include

• The twin prime conjecture
• The transcedence of e+pi.

To check these questions, you need to run through cases, where at each point you have to check where a computer program halts. This means you need to know infinitely many programs halt. For example, to know there are infinitely many twin primes, you need to show that the program that looks for twin primes starting at each found pair will halt on the next found pair. For the transcendence question, you have to run through all polynomials, calculate the roots, and show that eventually they are different from e+pi.

These questions are at the next level of the arithmetic heirarchy. Their computational formulation is again more intuitive--- they correspond to the halting problem for a computer which has access to the solution of the ordinary halting problem.

You can go up the arithmetic hierarchy, and the sentences which express the conjectures on the arithmetic hierarchy at any finite level are those of Peano Arithmetic.

There are those who believe that Peano Arithemtic is the proper foundations, and these arithemtically minded people will stop at the end of the arithemtic hierarchy. I suppose one could place Kronecker here:

• Leopold Kronecker: "God created the natural numbers, all else is the work of man."

To assume that the sentences on the arithmetic hierarchy are absolute, but no others, is a possible position. If you include axioms of induction on these statements, you get the theory of Peano Arithmetic, which has an ordinal complexity which is completely understood since Gentzen, and it is described by the ordinal epsilon-naught. Epsilon-naught is very concrete, but I have seen recent arguments that it might not be well founded! This is completely ridiculous to anyone who knows epsilon-naught, and the idea might strike future generations as equally silly as the idea that the number of sand grains in a sphere the size of Earth's orbit is infinite--- an idea explicitly refuted in "The Sand Reckoner" by Archimedes.

### Most believe discovered: Hyperarithmetic heirarchy

The hyperarithmetic hierarchy is often phrased in terms of second order arithmetic, but I prefer to state it computationally.

Suppose I give you all the solution to the halting problem at all the levels of the arithmetic hierarchy, and you concatenate them into one infinite CD-ROM which contains the solution to all of these simultaneously. Than the halting problem with this CD-ROM (the complete arithmetic-hierarchy halting oracle) defines a new halting problem--- the omega-th jump of 0 in recursion theory jargon, or just the omega-oracle.

You can iterate the oracles up the ordinal list, and produce ever more complex halting problems. You might believe this is meaningful for any ordinals which produce a tape.

There are various stopping points along the hyperarithmetic hierarchy, which are usually labelled by their second-order arithemtic version (which I don't know how to translate). These positions are not natural stopping points for anybody.

### Church Kleene ordinal

I am here. Everything less than this, I accept, everything beyond this, I consider objectively invented. The reason is that the Church-Kleene ordinal is the limit of all countable computable ordinals. This is the position of the computational foundations, and it was essentially the position of the Soviet school. People I would put here include

• Yuri Manin
• Paul Cohen

In the case of Paul Cohen, I am not sure. The ordinals below Church Kleene are all those that we can definitely represent on a computer, and work with, and any higher conception is suspect.

### First uncountable ordinal

If you make an axiomatic set theory with power set, you can define the union of all countable ordinals, and this is the first uncountable ordinal. Some people stop here, rejecting uncountable sets, like the set of real numbers, as inventions.

This is a very similar position to mine, held by people at the turn of the 20th century, who accepted countable infinity, but not uncountable infinity. Those who were here include many famous mathematicians

• Thorvald Skolem

Skolem's theorem was an attempt to convince mathematicians that mathematics was countable.

I should point out that the Church Kleene ordinal was not defined until the 1940s, so this was the closest position to the computational one available in the early half of the 20th century.

### Continuum

Most practically minded mathematicians stop here. They become wary of constructions like the set of all functions on the real line, since these spaces are too large for intuition to comfortably handle. There is no formal foundation school that stops at the continuum, it is just a place where people stop being comfortable in absoluteness of mathematical truth.

The continuum has questions which are known to be undecidable by methods which are persuasive that it is a vagueness in the set concept at this point, not in the axiom system.

### First Inaccessible Cardinal

This place is where most Platonists stop. Everything below this is described by ZFC. I think the most famous person here is:

• Saharon Shelah

I assume this is his platonic universe, since he say so explicitly in an intro to one of his more famous early papers. He might have changed his mind since.

### Infinitely many Woodin Cardinals

This is the place where people who like projective determinacy stop.

It is likely that determinacy advocates believe in the consistency of determinacy, and this gives them evidence for consistency of Woodin Cardinals (although their argument is somewhat theological sounding without the proper computational justification in terms of an impossibly sophisticated countable computable ordinal which serves as the proof theory for this)

This includes

• Hugh Woodin

### Possibly invented: Rank-into-Rank axioms

I copied this from the Wikipedia page, these are the largest large cardinals mathematicians have considered to date. This is probably where most logicians stop, but they are wary of possible contradiction.

These axioms are reflection axioms, they make the set-theoretic model self-simialar in complicated ways at large places. The structure of the models is enormously rich, and I have no intuition at all, as I barely know the definition (I just read it on Wiki).

### Invented: Reinhard Cardinal

This is the limit of nearly all practicing mathematicians, since these have been shown to be inconsistent, at least using the axiom of choice. Since most of the structure of set theory is made very elegant with choice, and the anti-choice arguments are not usually related to the Godel-style large-cardinal assumptions, people assume Reinhardt Cardinals are inconsistent.

I assume that nearly all working mathematicians consider Reinhardt Cardinals as imaginary entities, that they are invention, and an inconsistent invention at that.

### Definitely invented: Set of all sets

This level is the highest of all, in the traditional ordering, and this is where people started at the end of the 19th century. The intuitive set

• The set of all sets
• The ordinal limit of all ordinals

These ideas were shown to be inconsistent by Cantor, using a simple argument (consider the ordinal limit plus one, or the power set of the set of all sets). The paradoxes were popularized and sharpened by Russell, then resolved by Whitehead and Russell, Hilbert, Godel, and Zermelo, using axiomatic approaches that denied this object.

Everyone agrees this stuff is invented.

8@RonMaimon: While you give lots of examples, I still don't understand how you decide which parts of mathematics are invented and which are discovered. It seems to me that the rule is "things I'm comfortable with are discovered, things I'm not comfortable with are invented", I would never consider that to be objectively justifiable. Compare with my answer, in which I give the straightforward rule: "axioms and notations are inventions; consequences of those axioms are discovery". – Lie Ryan – 2013-02-26T12:41:59.433

6@LieRyan: Things that have an invariant computational description are discovered, things that are produced from idealizations that are beyond any computation, so that their properties can be changed in different models, these are invented. It's the computational foundations, and it's the only reasonably objective answer, at least since 1936 when computers became available. Your answer is not good, because computers are invariant to axiomatizations, all reasonable axiomatizations give the same notion of computer. So computers are discovered for sure, and I'm saying that's all. – Ron Maimon – 2013-02-27T12:20:50.043

2@RonMaimon: thanks for describing your rule, I think I can see where you're coming from, you're equating computability with discoverability, am I right? My answer come from a different perspective, which IMO is more general than yours. My answer comes from distinguishing between tools/axioms (inventions) and their usages/consequences (discoveries). I don't think it's an issue that Turing Machine and Lambda calculus gives the same notion of computer, just like it's not a an issue that you can use either InkJet or LaserJet to print the same images. – Lie Ryan – 2013-02-27T16:17:03.237

3@LieRyan: It's not an issue exactly, it's something which means that axioms are not the thing you are studying. The existence of computation, and its independence form axiomatizations, means that it really doesn't matter what the axiomatization is, that you are ultimately studying the properties of computation. I don't agree that all axioms are equally meaningful--- the axioms are useful inasmuch as they describe accurately the results of computations. You could make up false axiom systems, like adding "PA is not consistent" to PA, and then you get an axiom system which is not more than PA. – Ron Maimon – 2013-03-01T07:21:24.797

@RonMaimon: I never said that all axioms are or need to be useful or meaningful, just like there's no need for an invention to be useful. Axioms does not describe the result of computations; axioms describes a universe. What you can find in the universe are discoveries, the result of computations are discovered within the universe described by the axioms. There are useful universes and there are not useful universe, just as much that there is useful inventions and not-so-useful inventions. – Lie Ryan – 2013-03-01T07:32:11.887

@RonMaimon: Again, I understand where your answer is coming from, it is a completely different perspective from where my answer is coming from. The nature of the question seems to me does not exclude the possibility that there could be different meaningful yet conflicting ways to answer it. It really depends on how you define "invention" and "discoveries". – Lie Ryan – 2013-03-01T07:37:06.703

3@LieRyan: Ok, I sort of agree with you, but the issue I have is that the statement "mathematics is about axioms and deductions" is clearly true, but it doesn't explain how you select axiom systems for importance, or why different axiom systems end up being equivalent, or why the axiom systems naturally form a tower of increasing strength, indexed by countable computable ordinals which form the proof theory of these systems. I think these insights are more important, but at a more basic level, you are right, and I can't complain too much. Thanks for the comments. – Ron Maimon – 2013-03-02T03:01:03.790

1... but there is an issue with the word "universe". When you say "universe", and this universe is infinite, as it usually is, then this universe is really a model, and there is always ambiguity in constructing this model, there are many models which model the same system. This is always true when the model isn't finite, so it's something you must deal with. To say "axiom systems describe a universe" ultimately requires a way to construct the universe from the axiom system, and this is Godel's completeness theorem, and it gives a countable model to an axiom system. This is computation at heart. – Ron Maimon – 2013-03-02T03:03:44.537

FWIW, the term is hierarchy, not heirarchy. – Rudy Velthuis – 2018-12-27T06:44:51.867

I think 9306781264114085423 x 39204667242145673 is already a perfectly valid representation for a certain whole number and thus you already "discovered" that number completely without any calculation. If it really was anything to "discover" in the first place, that is. – COME FROM – 2013-08-02T04:50:34.100

@COMEFROM: ok then, if you take this point of view, the question I am asking is what is the value of this number modulo 10, the value modulo 100, and so on, in sequence. The question makes sense no matter what your philosophy, and the procedure of performing the multiplication gives you nontrivial information about the value, namely the precise decimal digit values. – Ron Maimon – 2013-08-02T16:45:45.853

@RonMaimon: I agree that there are many techniques to get all sorts of information about that particular number and different representations of the number reveal different information. For instance it wouldn't be clear that the number is divisible by 39204667242145673 if you had given us the decimal form only. It seems that you use the word "value" to refer to the decimal representation of the number. I find that confusing. – COME FROM – 2013-08-05T08:20:48.303

@COMEFROM: It can't possibly be confusing, I was just using it as an example of a finite procedure that gives an answer to a question, "what is the decimal value of this multiplication". It wasn't like I was using it for some other purpose than illustration. – Ron Maimon – 2013-08-06T21:11:29.313

@RonMaimon: Sorry, at first I thought you were talking about "discovering" the number. I guess you don't think the techniques to find out that decimal value are also "discovered", not "invented"? Just trying to understand what you really mean by "finite stuff" and all of it being "discovered"... – COME FROM – 2013-08-07T06:49:46.283

@COMEFROM: I mean you can't dispute that the answer is always going to come out the same, that if you draw the appropriate box and count rectangles, it will come out right, that someone can't come up to you and say "The answer is 17 in my philosophy" and somehow be also right. This is to distinguish it from saying "the number of points in a continuum is aleph-2" which a lot of people said (including, briefly Godel), but which is freely adjustible by forcing, so it not absolute. I am equating the mathematical notion of "absolute property" with the philosophical "discovered property" – Ron Maimon – 2013-08-10T00:38:30.980

The properties of halting computer programs are absolute, and so discovered. The computer programs which are nonhalting are also absolute in the philosophy I am taking, but only really because they are expected to be proved non-halting in sufficiently strong systems. The non-absolute properties begin with the first uncountable sets, and when you do set theory and get to the level of the set of real numbers, half of all the questions you ask are fungible and non-absolute, like "Is there a non-measurable set?" "Is the number of reals aleph-1? Aleph-2? Aleph-17?" "Is there a Suslin line?" – Ron Maimon – 2013-08-10T00:41:38.050

@RonMaimon: Ok, thanks for the clarification! I guess I see your point now. However, as there is a lot more to mathematics than just theorems (or correct answers to mathematical questions), and a lot of that other "stuff" -- definitions, techniques, notation etc., the decimal system and multiplication as an operation for instance -- I would consider mostly "invented" rather than "discovered". – COME FROM – 2013-08-12T09:10:22.537

@COMEFROM: Ok, sure. I am talking about the relation between numbers, the result of halting computations. This is discovered. The notation for describing the algorithm is invented, because it has a lot of arbitrary choices, although sub-parts, like clever tricks for reducing complexity, I would call "discovered", but since it's a vague human term, I don't care. I was using a precise term to answer the question. – Ron Maimon – 2013-08-14T11:11:33.983

"the set of all functions on the real line, since these spaces are too large for intuition to comfortably handle" - I don't think is quite true. In functional analysis one focuses on the set of Riemann integrable functions or Lebesgue integrable functions or measurable functions because things stay "well behaved," not because of the size of the space being difficult to grasp. – James Kingsbery – 2015-06-16T20:41:47.727

@JamesKingsbery: I was considering the space of arbitrary functions on R, so that there are no regularity properties. For example, the topological space of functions from [0,1] to [0,1] in the product topology. In this case, there are intuitions like "can't I just pick a random independent value at each x?" which fail in ZFC. Where people consider function spaces, they treat the function-spaces like classes. You don't actually use the set-theoretic properties, you just are using set notation. Nobody ever well-orders the space of tempered distributions, for example. – Ron Maimon – 2015-06-17T01:27:46.137

"Is it possible to get two different answers and consider both correct?" To lots of "postmoderns", probably yes. Or not. Or whatever. "Anything goes." And they still question why society can't solve its problems... – Rodrigo – 2015-11-06T18:26:33.473

24

This is only a partial answer:

As a mathematician, I have been asked this sort of question from time to time. Like most other mathematicians, I tend to sort of evade the question, because it's tricky. Usually, the question is put in the form, "Are you a platonist?"

The reference here is to Plato's eternal form that we are able to recognize, and that allows us to recognize the world around us (it is not obvious, afterall, that we should still be able to recognize an amputee as a human when we first see him or her, for example). When forced to continue, I usually respond "No."

I think the fundamental problem with Platonism is summed up in Brian Davies's paper, aptly titled "Let Platonism Die." I also add - if a mathematical 'discovery' hasn't yet been discovered, does it exist? A Platonist would say absolutely. An intuitionist would either say that it does not exist, or it exists only in the sense that some current or future mathematical system, devised and formulated vulgarly by humans, will lead to many more theorems - i.e. it exists only as an extension of what we have already created.

But ultimately, I don't think that this distinction is very important aside from the theistic or neural implications. A Platonist would say that when we recognize a triangle, for example, it is because we are recognizing the Form of a Triangle, some idealized, perfect, transcendental object. This makes a lot of sense, because Platonism obviously has at its roots Plato, who read much into the divine relationship between mathematics and the world espoused by Pythagoras.

As a final note, I should say that many well-known mathematicians lie on both sides of the fence. The most famous Platonist, I believe, is Roger Penrose, who is most famous for his creation of dozens of non-obvious tessellations and tilings.

1Nice answer. Since you named a platonist you could also name an intuitionist, for example V.I. Arnold who once wrote that "mathematics is the part of physics where experiments are cheap". :) – Michael – 2013-10-04T17:18:17.293

3OK, I try again: "I also add - if a mathematical 'discovery' hasn't yet been discovered, does it exist? A Platonist would say no. " Really? No? I though a platonist would say that it does exist before it has been discovered. – Lennart Regebro – 2011-06-07T20:47:10.560

1@Lennart! WHOOPS! Thank you. I'm changing it immediately. – davidlowryduda – 2011-06-07T20:47:57.800

11

I'm going to posit, admittedly without any research whatsoever about those who've preceded these thoughts, that an "invention" is a kind of "discovery," and that whether a thing qualifies as an invention is—yup, you saw it coming—subjective.

For example, we might say that the wheel was "invented" on grounds of (1) non-naturality (originality), and (2) intention. That is, prior to the wheel, circle-and-axle forms did not exist in nature, and so of course no one could apply it to the facilitation of movement. Furthermore, it's hard(er) to imagine someone carving a circle with a hole, then carving a spoke, then putting the two together, without having the intention of rolling the circle on the spoke, in mind. These circumstances give us cause to say that the wheel was "invented."

But, it's not impossible to imagine, either, that someone might have carved a circle with a hole for absolutely no reason to do with the concept of rolling, then happened to stick a stick in the hole (again, for no premeditated or relevant reason), and only then (or sometime later) realized its property of rolling. Note how in this case, we're more inclined to call the wheel a "discovery!"

I think we tend to call novel discoveries with premeditated results, "inventions."

So, I would say mathematics, as a general notational/deductive system, was mostly invented. But its concepts were discovered. (And even some notations were indeed discovered, while striving for convenience, concision, and pictorialization!)

11

I think the words "invention" and "discovery" are a bit poor to describe the birth of mathematic if there is one. It makes no sense to me to say mathematic has appeared as when Christophe Colomb discovered America or was invented as the boomerang.

The word mathematics might have been invented, the language in which the mathematics are written might have been invented but the abstraction movement from the real word, the structured synthesis that it undertakes, all that give thickness to mathematics themselves (it depends what you call mathematics) are part of mankind. You don't ask if beauty has been discovered or invented ?

My personnal point of view is that the question "what is mathematics" would be more serious, I would found even more interesting "why do we do mathematics".

9

Both.

Formal mathematics is created by people, and doesn't necessarily relate to anything in our world.

However, the history and progress of mathematics is many times related to applied mathematics, which is related to our physical world.

In other words - geometry will remain valid even if we will find out that it doesn't hold true for our physical world (and actually, it doesn't...) - But it's hard to believe many people would have started researching this field as a pure abstract field, with no relevance to real problems of construction, navigation, etc.

8

Mathematics is an abstraction. As such it is invented by humans to deal with concrete things is a more practical manner, by giving us generic tools to deal with the specific.

Later more mathematics was invented to deal with the abstractions of earlier maths, leading to more and more complex abstractions, but the invention of math was done to deal with concrete things, like geometry and trade.

2

Mathematics is also highly related to aesthetic concerns, which don't necessarily have to do with concrete things. Music is highly mathematical, and there are many other examples as well, so I don't think it's accurate to view mathematics as entirely an abstraction about concrete things.

– eMansipater – 2011-06-07T21:39:52.847

3Topics in advanced number theory, non-euclidean geometry, Kolmogorov complexity and many other branches of mathematics were certainly not invented as a way to "deal with concrete things in a more practical manner." – Ami – 2011-06-07T20:22:24.210

@eMansipater: 1. Music is highly concrete. 2. It was about the invention of mathematics. It was invented to deal with concrete things. – Lennart Regebro – 2011-06-07T21:49:47.323

@Lennart but the aesthetic concerns with regards to music aren't, which was my point. And regarding 2, that's only if it was invented. What if mathematics is as much a "thing" as anything else is, and was discovered rather than invented? Since this is the original question, the main point of my comment was merely to note that your reasoning is probably insufficient to argue that mathematics was invented; though if we assume it was it might offer a plausible explanation. See the difference? – eMansipater – 2011-06-07T21:55:11.053

@eMansipater: Yes, I see, and you are wrong. See answer above. :-) – Lennart Regebro – 2011-06-07T21:57:54.890

2@Ami: Fair enough, some of them where invented to deal with abstract things (namely other maths) in a more practical matter. But those abstractions are done to handle other abstractions which in the end are there to deal with concrete and complex matters. (Non-euclidean geometry does have many extremely directly practical applications though. Maps of earth, for example). – Lennart Regebro – 2011-06-07T20:28:39.547

1-1: For reasons explained by eMansipater and Ami. – Q__ – 2011-06-11T17:57:25.983

@Lovre: They didn't explain anything, but you are welcome to do so if you want. Yes, math can be used to handle abstract things. I never contradicted this. But it wasn't invented to do this (because those abstractions in turn was invented after mathematics was invented). Quite simple, really. – Lennart Regebro – 2011-06-11T18:41:41.853

7

Mathematics is a lot of things: there are basic/complex entities/structures, proof strategies, algorithms, formal manipulations... in order to try to answer your question I think we should make some distinctions between different matematical entities/activities where the "creative" part of the thought is more or less relevant. Moreover some parts of mathematics seems to be neither discovered nor created, they seem to be just "given" embedded in our natural language grammar.

Some examples of math entities/activities that:

• seem embedded in our grammar: classical logical operators, classical deduction rules, tautologies, natural numbers
• seem more discovered: non-trivial general fact in a given structure (ex. fermat's last theorem), finding general patterns, classifications, finding counterexamples
• seem more invented: definition of new non-trivial structures (ex. complex numbers, quaternions), finding new non-trivial proof strategies.

6

First, Quine: "..[If externally true] the definitions [of mathematical laws] would generate all the concepts from clear and distinct ideas, and the proofs would generate all the theorems from self-evident truths." "...the truths of logic are all obvious or at least potentially obvious..[but] mathematics reduces only to set theory and not to logic proper." -Epistemology Naturalized; Chapter 39.

The implications are bleak for the ontological objectivity of mathematics. For a fact to reduce to certainty one must present sensory evidence (to be "self evident"). Consider, I see that things fall to earth and stay there. I explain this to myself with physics. What I see is not physics. Physics is a framework invented to generalize what I am perceiving.

A 1 and a 1 on a sheet of paper are not the same as a 2 on a sheet of paper. 1 is the smallest prime#, for example, while 2 is the smallest even prime, among myriad other differences.

An apple on a table and an apple on a table is not the same as two apples on a table, as the set of two apples could be different apples. I cannot cube two apples, except to make pie. But I cannot make pi with an apple.

The value of a dollar is measured mathematically. But if humans disappear, the piece of paper remains, while the value disappears with humans. Things stick to the earth regardless of our existence, but the theory describing our perception of gravity does not.

The epistemic objectivity of Mathematics is ontologically subjective. It exists only in our minds. Something that exists only in our minds can only have come into existence within our minds. Something that does that is invented.

No, the "value of a dollar" is not in the least "measured mathematically". And your other examples don't relate to mathematics either, as far as I can see. – Ingo – 2018-08-25T10:17:50.323

Discoveries exist only in your minds. Do you say that all discoveries are inventions? – Little Alien – 2016-08-04T15:03:30.607

5

This is a serious question and it is the same as saying: is the knowledge in mathematics universal or a human construct?

Pi (the number, regardless of its base) and many other things are universals, mathematics are discovered to that extent. Then they can be used to formalise inventions that may prove to be wrong, right or paradoxical, in the same way the knowledge (discovered) about horses and rhinos can be used to (invent and) speak about unicorns (that were never discovered).

Can we say (as many answers point here) that biology was invented because of unicorns?

I like this answer. Humans invented the wrong stuff, the rest is discovered. Apart from being funny, I think you've nailed it :-) – user2808054 – 2014-08-04T15:02:39.010

But constants like Pi are only a function of euclidean geometry. It represents a consistent ratio between circumference and radius, but only in 2D. In non-euclidian geometry, the constant fluctuates based on the altitude of the center versus the circumference. Therefore, despite being a seeming constant mathematical law, it is really just a relational coincidence. This sort of understanding demonstrates how despite "discovering" a mathematical constant or rule inductively, it does not imply some inherent quality. Instead it is merely a tool invented to represent a repeating pattern. – PV22 – 2016-06-14T06:24:29.813

However, the most fundamental mathematical terms are defined a priori. i.e. "1" can be represented multiple ways, but it is always the same rudimentary concept. Whether shown by... Roman Numeral: "I" Hebrew: "Aleph" Hindo-Arabic: "1" Putting 1 apple in the basket Tapping once on a table The basic premise of "1" is understood, and transcends specific semiotics. If you can differentiate one thing from another to any degree, then you can grasp the basic premise of numerals, and therefore mathematics. The discovery is that first realization that things can be differentiated from each other. – PV22 – 2016-06-14T06:34:55.600

4

If by "was it discovered?" you mean "was it there all along?," I think the answer is "yes." Consider that we can use math to "predict" the past ("retrodiction"). A similar concept is "hindcasting," where the validity of a scientific model is tested against data that was recorded before the model was even invented. Presumably, in order for retrodiction/hindcasting to work, the mathematics had to be there all along, constraining the evolution of the universe. If you buy this argument, this suggests that mathematics was there all along, or "discovered."

Of course, other definitions are possible.

3

My view on it is that Mathematics is a system invented by humans to represent things we otherwise can or cannot perceive. For example, we can perceive an object through vision and know it's a triangle, however, our vision alone does not tell us the length of the legs of the triangle. We need math to represent that for us.

JUst to further my point, consider Calculus. Two people who were on completely different sides of Europe, Leibniz and Newton, created a system that that both do the same thing. For Newton, f'(x) is the same as Leibniz' df/dx. Both of them yield a function that represents the slope at any given point on the original function, f(x). They invented a system to represent something we otherwise couldn't perceive (which was pre existing - The shape of a mountain should be enough to prove that the slope exists naturally), the only difference was their notation.

Newton didn't use function notation (f'(x)). Newton used geometric demonstrations in his Principia. (Look at "Lemma 1" in the Principia.) He built on Archimedes's method of exhaustion (a method that Galileo used as well, think of his infinitely sided polygon, i.e. the circle). Whereas, it might be said that Leibniz's notation came about through his interest in summing infinite series. His approach is radically more concise than Newton's--but Newton purposefully didn't use a different form of notation (from that of Apollonius) because he didn't like how ungrounded calculus was. – Jon – 2011-11-08T20:23:28.300

Telling that they invented a thing that was pre-existing is self-contradictory. You are asked if it is invented or discovered with the difference defined as discovery points to pre-existing things. You point to pre-existing things and say that this means that we have an invention rather than discovery. Aliens cannot understand this logic. – Little Alien – 2016-08-04T15:10:00.357

I did not say they invented a thing that was pre-existing. I said they invented a SYSTEM to represent something that pre-exists. – MGZero – 2016-08-04T15:41:51.730

3

Discovered, if it was invented, then whoever came up with π in theory could have just made it equal 3, but instead they discovered it, and that it was an irrational number. The math was discovered, but the different techniques and conventions used for calculation were invented. Kind of like, Physics; the laws of physics already existed, but man has discovered how to use them to their advantage with their inventions.

Math and phisics is invented. You contradict yourself when say that techniques are invented, that they don't exist before opened, but MathoPhyscs is somehow special. You say that apples are discovered but peers are invented. This is not fair and self-contradictory. The (shortest) route from A to B did preexist as the connected locations. There can be many ways from A to B but saying that they are invented cos they did not pre-exist just because there are many is groundless. I can show you a landscape with multiple paths which do exist before you 'invent' them. – Little Alien – 2016-08-09T07:27:00.867

The Math itself is nothing more than a bunch of techniques that we pre-invent (or discover) for our daily use. So, telling that math is discovered whereas techniques are not is self-contradictory, no matter how you look at it. – Little Alien – 2016-08-09T07:28:03.820

3

Mathematics is normative. That is clear when one reads Euclid and Lobachevsky in juxtaposition, or Euclid and Descartes, or Euclid and Leibniz or Newton, or Leibniz and Newton and Dedekind, or Dedekind and Canton, or Canton and Godel, etc., etc.. Geometry is clearly normative, as we have different geometries (although one might claim, "yes, but they can all be transformed into one another"). But the argument goes like this: there is no other arithmetic; and thus, in counting (and its extensions), we are discovering something fundamental to the universe. Of course, such an answer supposes that Euclid and Dedekind are talking about the same arithmetic. Is that even possible? No. There's no room, in Euclid's conception of number (think of Books V and VI of the Elements), for Dedekind's cuts, and thus, no room for a whole host of numbers that are incompatible with Euclid's concept of number. And if you think that the concept of number is fundamental to a conception of arithmetic, then it would seem that every time we "add" new "sorts" of numbers (which are invented by new sorts of functions), we create a new arithmetic. But, someone might say, "that's all well and good, but we really just subsume those other arithmetics under what we call arithmetic--there's really just one arithmetic." But that would be like saying "wave-mechanics really just subsumed ordinary mechanics...." Such a statement doesn't make any sense.

3

I think it's hard to say. If you believe that mathematics has been discovered, you must assume that "something" is out there, something we can interact with, of which we have been unable to prove existence so far.

However, even assuming that there are ideas out there, I believe that there is no reason to think that humans should be, in any way, able to understand them. As David Deutsch famously said, the fact that we understand the laws of Nature, is pretty much like saying that you land on another planet, and find aliens completely able to speak to you in english.

Last but not least, it is possible that our models of how the Universe works are completely wrong. Hence, we are talking about ideas derived from our models that may be, ultimately, way off the truth.

3

I think the distinction between discovered and invented is mostly about how one chooses to define these words. My personal definition would be that when you can reasonably assume that many other people can in principle find the same thing X, then X can reasonably be said to be discovered, but when X is pretty arbitrary, like a particular notation, then it's invented. For example, different people can discover the Mandelbrot set, and various relationships and figures in there:

In the above image the colors are an invention, not a discovery. Different people will maybe choose similar coloring here, but I think it's pretty much an artistic choice. The colors roughly reflect how fast a point in the complex plane will head off to infinity under a certain repeated square-and-add operation, but they depend on a lot of parameters (including how many iterations one deems sufficient to establish the wayward nature of a point), including, of course, some particular color palette.

I think this illustrates nicely that the very same mathematical beast can have aspects that are discovered, and aspects that are invented. ;-)

3

In line with many others' probing at just what 'invented' means, invention and discovery can be viewed as the same thing, as both require the application of a set of steps along with various objects under consideration. Even when discovering, say, a continent, the notions of continent-ness and America-ness are both inventions, nonetheless. And even when inventing, say, the internal combustion engine, the laws of physics which allowed such a device to exist were in place before the invention, and thus the particular arrangement of parts which effects its existence was discovered.

3

If only we would get the question right, we may be able to get the right answer. The problem is, is invention discovery or creation? As a seven times patented inventor, I will tell you that invention is, at least to a great extent, discovery. As my patent agent explained, what is invented is a "method", a way of getting a job done. During the process of invention, one tries on a gazillion methods of getting the job done that don't work. When one does discover a method that does work, well, one has an invention.

The proof of discovery verses creation, is the proof of reproduction. When a person who has never seen a wheel before tries to solve the problem of causing heavy objects to move, he may very well re-invent the wheel. This happens all of the time with inventions. One comes up with a method of solving a problem, only to discover that someone else has patented that invention before him. Creativity is not like this at all. If two people truly independently come up with the same creative product, then their creative product is, well, simple. In fact programs are used to analyze college papers for plagiarization. They seek matches in a 7 word sequence because it is unlikely that two people independently come up with seven little words strung together the same way.

So let the question be, "is mathematics discovery or creation?" Ask the anthropologist to seek out the mathematical methods of other cultures. Surely these methods would be extreme subsets of our math. However, they still have some simple consistencies. Two plus two (though represented with different words) equals four. The fact that two cultures independently come up with the same logic sets establishes that mathematics is discovery, not creation.

I like your reasoning here, but it only works for very simple kinds of mathematics. When we start getting into areas like trigonometry and real analysis, it's plainly not right to say that different cultures came up with these things independently - rather, methods moved from country to country as people from different cultures traded, fought and explored with each other. – Paul Ross – 2013-09-11T06:37:36.633

3

A little bit of both. One invents the mathematical concepts, and then discovers the consequences of these concepts. Something like "define lines and points via axioms, and then discover triangle properties."

Then one wish for different consequences and invents new concepts, something like "I wish triangle had sum of angles over 180 degrees; let's define lines as great circles on the sphere instead of lines on a plane and see what happens."

And it goes on and on, invention hand in hand with discovery.

3

My elementary math lecturer likes to say

God created the number 0, and the successor. The rest was invented by mankind.

I think there is some truth in this quote, even if you don't believe in God. So to answer your question: I'd say that the very basis of math was discovered, but most of the sophisticated math was invented.

2

The Black-Scholes equation describes the price of a stock option over time. Since the concept of stock options, financial markets et cetera were invented, not discovered by humans, does that suffice as an arguement that mathematics was invented? If there was no such thing as a stock option, there almost certainly wont be the black-scholes equation. The black-scholes equation would never be out there waiting for us to discover it if there was no such things as a stock option.

If one claims that although a stock option was invented, the black-scholes equation can be said to be discovered, how many more mathematical theorems, equations, models and so forth are out there that are waiting to be discovered, dependent on our future "inventions and creations"?

2

Mathematics was invented as a means of expressing numbers, relations, etc. while the laws of mathematics were discovered.

Pi is Pi, whether you like it or not. It's value was discovered. However, expressing Pi in base-10 decimal notation as 3.14* (or 22/7 if you're that type of person) is an invention of the human mind, whereas the actual ratio was such from the beginning of time.

In short, mathematics is a human invention to better understand and to discover how the natural world operates on a purely logical level. One needs to separate the method from the observed.

2

Math is a system made up to quantify, measure, understand, and determine things by mathematical proof, logic, analytical reasoning, and common understandings. It's also an abstraction as well since the actual theoretical basis given on the implementation of the something will usually differ in practice atomically, etc.

Math is an endless study of conjectures that is agreed upon by people subscribed to such a phenomenon. Math has been used for centuries to keep track of things, measure qualities of things, and in modern days to analyze and interpret highly complex conjectures, theories, and explanations of everything around us.

Was it invented or discovered? Philosophically speaking, is anything ever really measured or discovered?

Somethings just are, and to our best knowledge we have a system, math, to quantify and analyze things.

Math never "was" anything until it was agreed upon, brought in to use, and implemented, agreed on, and understood. Such highly complex systems never were used by the biological creatures way before us, e.g. fish, bacteria. Quantity is just mass without numbers, and quality is just coincidence without observation.

A response to another question I found here that spiked my interest:

why does the concept of number exist, but the concept of complex number invented?

The concept of everything tangible and/or intangible only exists to be understood based on the reality and observation of the phenomenon around it, how those phenomena perceive it, agrees upon understanding it, and how well that system can accurately model the underlying reality of it. To a human a ball is something you kick, throw, catch, has shape, mass, volume; to a dog it's something in its way. The reality is that if there's a reality under the underlying concepts we try to figure, only such a system invented will try to emulate the process of understanding it more and more.

The question also touches on the grounds of everything around us, and its totality. Let me give you an idea of why I am proposing that math is an invention:

Before people could even count, or even existed there were always many different biological structures, masses, gases, inanimate objects, and collective existences outside of a singular model, single perception of electromagnetic radiation's visible light, eyeballs, brains, or classification itself. Before we evolved did dinosaurs, assuming you believe they existed, count and classify the world around them? Probably to a mere, limited extent, but not anywhere near how most people would think of it. All biological creatures that have evolved past the bacteria have gained perception, analyical minds, and the capability of complex thinking to become better suited to the existence to around them. None of them ever came anywhere close to modern humans.

I doubt the fish in the sea can accurately model multiple perceptions of visible light on masses, and use their brains to visualize this as two separate objects, thus, manipulating the abstraction of items, beings, or existences around them. However, we look at two things and agree these are two things. We see two rubber balls on the floor, and we come to the immediate conclusion that they are two distinct objects. But are they really two things, or have you just subscribed to a common method to segregate objects based on human evolved, educated, or brain limited rules?

Point is, you see two non-connective items, and you classify/label them as two. You aren't, in most cases, visualizing the ball as a synthetic base of polymers, isoprene, and other chemical elements and masses that constitute its existence within electromagnetic radiation in an atmosphere. Therefore, you have classified the existence of two balls based on segregating instances of light, however, you are only using a system to do so that is 100% limited to your brain's understanding.

Without a system, understanding, or method of perception everything would exist, but would not be calculated, observed, or manipulated.

1

I consider an answer too simple if it just affirms one of the alternatives and negates the other.

Naming just a few eminent contributions to mathematics: Complex numbers, set theory, theory of schemes. E.g., the concept of a set has been invented by Cantor, it did not exists before. After the basic concepts like set, power set, cardinality etc. had been invented, the Continuum Problem was discovered, hidden deep in these concepts.

Therefore I compare mathematics to a game like chess: Inventing new mathematical concepts is like creating new rules of the game. Playing a match means to discover the consequences of the rules and to solve the problems posed by the rules.

My conclusion: The rules of the game of mathematics have been invented. Following the rules mathematicians then discover some challenging matches.

1

Neither, it is understood. You are mathematics, everything you experience is mathematics, everything you think you know is mathematics. Your brain is an intricately connected computation machine which gives rise to all of your experiences and to your sense of self. Mathematics is the ability to predict the future; it is the ability to remember the past.

2Very beautifully put, but I disagree, and as this is nothing but your opinion (as it currently stands), we have to agree to disagree. – iphigenie – 2014-01-17T18:12:19.973

@iphigenie It is a statement of fact. Consider the Cybenko theorem which proves that a rather simple artificial neural network, a computer coded approximation to the way our brains function, has the ability to reproduce any continuous function in $\mathcal{R}^n$.

– Chris Mueller – 2014-01-17T20:09:36.040

It is not a fact, otherwise there would not be an ongoing philosophical debate about it. Welcome to philosophy.se, you can't just claim to be stating objective facts. – iphigenie – 2014-01-18T11:03:56.660

@iphigenie Lol, this is your definition of a fact? What if I told you that electromagnetic waves travel at the speed of light; is this simply opinion because the philosophers aren't debating it? It truly is a fact that your brain is a computation machine. It truly is a fact that mathematics, through natural law, allows us to predict the future. It truly is a fact that modern digital memory is based on the field of binary mathematics. It may be my opinion that "everything you experience is mathematics, everything you think you know is mathematics", but I don't think it is such a far leap. – Chris Mueller – 2014-01-18T15:04:00.120

The brain might be a computation machine, among other things, but that doesn't automatically mean that it "gives rise to all of your experiences and to your sense of self". THAT is not a fact, and, unlike your other "facts", this is a philosophical claim, which isn't undisputed. That's my point. – iphigenie – 2014-01-18T15:07:00.787

@iphigenie I'm not sure that philosophers have the proper toolset to address the physical origin of one's sense of self. Certainly not without consulting the natural sciences. Btw, thanks for the spiteful downvote. – Chris Mueller – 2014-01-19T16:42:00.063

You're not sure, and you provided not one single reference, but you still claiming it. My downvote needs no further justification. Btw, no need for personal grudge, and you had no evidence at all that is was mine. – iphigenie – 2014-01-19T19:40:50.610

@Chris, at a less abstract level consider: "You are genetics, everything you experience is genetics and physiology, everything you think you know is neurophysiology. Your brain is an intricately connected computation machine which gives rise to all of your experiences and to your sense of self. Genetics is the future." I long for the days when humanity will rebuild itself into the "Oh so media present" top one-percenters. But, I would settle for the productive forty-percenters. Mathematics will get us there, but only through biocomputation and genetics. – Darcy Davis – 2014-11-13T21:29:36.837

0

From a Neo-Intuitionist perspective, to the degree mathematics is invented, it is still discovered.

Did we invent, or discover the consonant 't'? We discovered that our mouths reasonably make that sound, across a wide swath of our species. But we decided that this was an important thing, and in so doing, we invented the idea of 't'. We invented a consonant by discovering a fact about ourselves.

From this perspective, mathematics is a set of ideas to which humans are naturally attracted in a given way. But those ideas themselves are a product of the human mind, the way the consonant 't' is a natural product of the human vocal apparatus. Those ideas arise out of individual humans, who can be considered to invent them. (Someone first uttered the sound of t. Someone first asked if -1 has a square root, or whether infinity comes in various sizes.)

But mathematics chooses out the ones that feel a given way and isolates those that appeal broadly to a given emotional reaction. In that sense it is a branch of psychology which discovers things about human thought.

It elaborates those ideas to a degree that makes it seem like it is creating things, but really, it is exploring our shared fund of ideas for ones that seem purely symbolic and not worthy of questioning, and sees how their consequences fit together.

0

This is an observatiin I can't remember where I heard, so would be greatly obliged if anyone else knows. But I think it's a killer line of argument.

Consider that somewhere in the set of all rational numbers, is the answer to any question you could ask (taking the numbers as eg ASCII codes). Yet knowing this does not give you these answers. It would take ennumeration of a number, and then a relational process to check it and confirm it is correct.

So by this model, ennumeration, and checking relations, are not magically external to the properties of a number, but fundamental to it. Invented not discovered, QED.

-1

An invention is something that did not previously exist. A discovery is made of something that already existed. Therefore, mathematics was invented as it did not exist before someone created it. For example, the number 1 exists only insofar as imagined and not in nature.

“One of the principal objects of theoretical research is to find the point of view from which the subject appears in the greatest simplicity” (J.W. Gibbs). — This should also be aimed at in philosophy, as I tried to suggest here.

– None – 2017-06-29T16:01:08.953

2This doesn't really add anything to the existing answers (for example, the top voted one), and is nothing more than providing an opinion without any references. Please read the [help] to learn what kind of answers we're looking for here. – None – 2015-06-16T18:55:48.533

This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – James Kingsbery – 2015-06-16T20:12:11.770

1@JamesKingsbery The question is "Was mathematics invented or discovered?" Can you explain why my answer is, in fact, not an answer? – Ronnie Royston – 2015-06-16T20:22:29.530

1@Keelan My answer is more simple than previous ones and is easier to understand and clearly provides an answer. The only possible reference is to a dictionary which would be condescending. Is there no value in simplicity? – Ronnie Royston – 2015-06-16T20:25:42.850

1It is more simple, and doesn't add anything. – None – 2015-06-16T20:28:32.467

1@Keelan Is there no value in simplicity? – Ronnie Royston – 2015-06-16T20:29:03.600

Not in this manifestation, and in general not when it comes without references. But anyway, I don't think we're going to agree on this. James and I left a flag on this post, so a moderator will pass by and he will decide what to do. – None – 2015-06-16T20:30:48.667

1@RonRoyston, I'm very much in favor of simplicity. Your answer is begging the question though: "Therefore, mathematics was invented as it did not exist before someone created it." - the matter at hand is whether it existed before any humans thought of it, and you've assumed that to be the case and said therefore it was invented. – James Kingsbery – 2015-06-16T20:32:44.050

1So, what in this manifestation negates the value of simplicity? I am eager to learn. Can you suggest what I might add in terms of references? – Ronnie Royston – 2015-06-16T20:32:58.370

1By way of constructive criticism: I think a good answer to this question should provide a rational basis for why mathematics was invented or discovered (or which parts of it fall under which category). The top voted answer is a very good example to follow: it is very simple, but describes two different common rational bases for deciding the question. – James Kingsbery – 2015-06-16T20:38:36.883

1@James, I did not assume math existed prior to human thought. I said that math did not exist prior to human thought and gave the example that the number one (a concept) came into existence only when thought of by man. – Ronnie Royston – 2015-06-16T20:40:59.037

3You and your simplicity. Let me answer you with a quote of a smart person, too (Wittgenstein): "Some philosophers (or whatever you like to call them), suffer from what may be called ‘loss of problems’. Then everything seems quite simple to them, no deep problems seem to exist any more, the world becomes broad and flat and loses all depth, and what they write becomes immeasurably shallow and trivial." Please stop spamming what you consider "simple" answers and start abiding by the rules of this site. – iphigenie – 2015-06-16T22:03:07.403

1@iphigenie I am open to learning. What is lost in my answer above? How is it spam? – Ronnie Royston – 2015-06-16T23:47:19.243

3@RonRoyston Take it to meta. – iphigenie – 2015-06-18T20:54:52.837