Epistemologically is mathematics considered more of an invention, or a discovery?

I know it’s a fairly trivial question, but I am curious. Epistemologically, is the origin of the human use of mathematics considered more of an “invention” or a “discovery”?

in-vent (in vent’) v.t. <-vent-ed, -vent-ing>
1. to originate as a product of one’s own
ingenuity, experimentation, or
contrivance: to invent a better mousetrap.

dis-cov-er (di skuv’uhr) v.t. <-ered, -er-ing>
1. to gain sight or knowledge of (something
previously unseen or unknown).
2. to notice or realize.
3. Archaic. to make known; reveal; disclose.

From what I’ve seen, support for both sides is pretty much equal, and there’s no particular piece of evidence that can’t support both sides equally well. So it’s really a matter of which notion you like better.

The “discovered” folks are known as platonists, and the “invented” folks are known as formalists. This was a big debate at the beginning of the 20th century, and now you don’t hear much about it any more. They tell me something called structuralism is popular these days, but I’ve never seen a particularly intelligible exposition of it.

For what it’s worth, a mathematician’s opinion on this question doesn’t have any bearing on his/her ability to do mathematics.

Um, can I vote for both? As I understand it, mathmatics consists of a set of definitions of various concepts, and rules for determining things about those concepts (the rules of logic, etc.), and then a whole bunch of theorems that were discovered using these definitions and rules. (Perhaps there’s a bit more to math than definitions, axioms, and theorems – algorithms, for instance – but the claim that an algorithm works is a theorem, or at least a proposition. So I suspect most of math fits into these three categories.) You can’t really say a theorem was “invented”, because the theorem was already true given the definitions and rules, even before you suspected it was true, much less proved it was true. As for the basic definitions and rules, it might be reasonable to say these were invented, but it was “discovered” that these definitions are useful in dealing with a wide variety of problems.

So I guess if I had to pick one or the other I’d say it’s a little more discovery than invention.

Was the Wright brothers flirst flight an invention or a discovery? Well, it certainly involved the invention of a machine that could fly, but in building this machine they did a lot of experimentation, and discovered a lot of facts about how such machines operate, e.g. facts about aerodynamics. A lot of things (including mathematics) are both inventions and discoveries.

In one of his lectures, Feynman asks himself why math is so effective for understanding physical phenomena. His answer is that math is basically pre-packaged logic which can be applied to anything that meets the assumptions of the logic being applied. One can use this to make valid conclusions, and know what are consistent or inconsistent results. This cannot be done intuitively in many ostensibly obvious situations.

What I got from that is that the material universe is a logical place. Gaining sight or knowledge of the fact that the universe is logical, noticing and realizing inferences that are logical, and making known logical relationships, regardless of whether there is a physical entity to which some logical relationships can be applied, would seem to be more in the realm of discovery than invention.

It seems that to say that math is an invention is essentially saying that we are imposing physical constraints on the universe that previously didn’t exist. For example, things arriving randomly over set periods of time follow a certain probability distribution, the Poisson distribution (IIRC). If I want to supply my restaurant so that 80% of the days it is open I will have enough staff, I can keep track of how many people arrive each day for a while and then plug in those data and get my estimate. If Poisson had never been born, if probability and statistics had never been developed, would the arrival rate of customers to my restaurant be something completely different? Would my customers show up in patterns matching no discernable distribution? Before probability was known, did rolls of fair dice produce results where some number happened very frequently relative to the rest? Before calculus, did an arrow shot from a bow reach its maximum height at some place other than where its vertical velocity was equal to zero?

Benford’s Law didn’t cause the pages of logarithm (sp?) tables to become more worn at the low numbers than at the high numbers. That happened without anybody knowing about it, or thinking about why such would be the case. Benford noticed an extant phenomena and set out to make sense of it and the result, Benford’s Law, makes known a phenomenon that is a product of living in a logical universe.

I guess that’s more of a GD answer than a GQ answer…but, hey, that’s what I get when I’m cyber-slacking.

Thanks to everyone for replies!


vBulletin should have a facility where a poster can automatically copy-paste an existing post by referencing that post number in some special tag.


astro, this book by Lakoff is dedicated to answering your question: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.

I would argue that the notations that we use for mathematics are invented – they were made to represent our mathematics, and in some cases, more than one invented choice was available.

However, the relationships that are found using these notations are discoveries – we “discover” the formula for an ellipse, or that the ratio of the circumference of a circle to its diameter is a constant.

The funny thing is that, while one might expect the invention (notation) to happen first, followed by the discoveries, many of the discoveries happen first, after which we invent a notation we can express it with. We invented a symbol to use for pi after we discovered that it was a constant ratio – because before then, it wasn’t obvious it was needed. This then gives us a foundation to discover more.

I think the reason this question is hard to answer is because mathematics does advance in cycles: discover, invent, discover.

Another vote for “both”. That is, the logical truths of mathematics are discovered; what was invented was a series of tools for thinking about math in a useful way that could be taught to others. It’s sort of like the debate over whether mental concepts would exist if we didn’t have the language to express them.

Platonists do go for “discovered” and formalists generally go for “invented”, but it’s really overstating it to identify them. Basically, to answer the OP, you’ll find most mathematicians have a general feeling and can answer the question if put on point, but in their hearts of hearts (how do you pluralize “heart of heart”?) they don’t see it as mattering all that much in how they do their work on a day-to-day level. Philosophers no longer generally have the mathematical chops to discuss any mathematics past the 19th century, so very few people bother with philosophy of mathematics. If the workers themselves don’t care and the philosophers can’t handle it, the question goes unanswered.

To answer everyone who’s trying to split the difference, the question is better restated very similarly to something you just said, Lumpy: “Do mathematical truths exist a priori – independant of anyone thinking of them?”

Consider the Twin Prime Conjecture: There are an infinite number of pairs of prime numbers differing by exactly two. This is a statement about the ring Z of integers. Now, is this statement true or false independantly of someone being able to prove it (in which case a proof or disproof would be a discovery), or is its truth or falsity determined by the existance of a proof or disproof (in which case such would be an invention). You can’t have it both ways.

As to the “concepts vs. tools” idea, it falls apart on closer scrutiny since the tools are logic, which itself falls under mathematics. Was the principle of mathematical induction invented? You might think it’s a tool rather than a concept, but it’s really a statement about the Peano characterization of N: that all natural numbers are either 0 or the successor of a natural number. If something is true of zero and of the successor to any natural number, it’s true for all elements of N. The fact that this (along with a couple other rules) uniquely characterizes N is a theorem, and thus a concept rather than just a tool.

This is not strictly an epistemological question. Epistemology concerns itself with the limits and validity of knowledge. (How do we know? How much can we know?) Metaphysics, or specifically ontology, concerns itself with the nature of being or existence. (What is it?) As you can see, the two can overlap in some ways (how do we know what it is?), and certainly our answers to some questions will affect our answers to others; this has been most recently considered an ontological commitment, that is, by accepting the truth of various propositions, we commit ourselves to a specific ontology.

I have never seen clear presentations of the philosophy of mathematics except in the two ways ultrafilter mentions, though whether that is a testament to the strength of those two positions or simply my lack of exposure is not something I’m willing to state one way or another.

It seems, though, that if one remains committed to the notion that numbers exist, then one will probably side with platonism (anti-realism in mathematics is hard to come by without denying the truth of a great number of things we generally consider true), and as such one would generally be compelled to suggest that mathematics is a discovery of abstract objects, relationships, or most likely both.

Bertrand Russel, for example, commented in The History of Western Philosophy, “…[N]umbers are, in a certain sense, formal.” (I believe this is a direct quote, though I can hunt it down if anyone desires.) This speaks of a formalist view (whether Russell was a formalist is not really important here), which is roughly the position that mathematics is simply a complex set of rules for symbolic manipulation; that is, one could say, a game. This would make mathematics an invention.

Historically, mathematics has come about from both paths; that is, we have “mathematicized” observations (much of scientific activity involves divining the relationship implied by data), and we have also chosen pre-existing abstract mathematics to fit observations (non-Euclidean geometries came about before the application of those geometries to the problem of space-time, for example); and it is fair to say that some mathematical behavior fails to fall into either camp. (The calculus, for example: was this in response to data, or a way to explain the data? – the answer almost seems to be “both.”) To me, this does not indicate that platonism or formalism are somehow necessarily incomplete or wrong, only that the question of mathematical objects’ existence is probably at a level of discourse prior to the collection of evidence (since, frankly, it is something that we use to characterize the collection of evidence). Considerable weight for this position comes from the argument that mathematical knowledge is a priori, which means that it is independent of experience. Roughly, it would be suggested that no matter how many times I put one liter of water in with another liter of water and get two liters of water I would never come to the notion that “1+1=2”. I think there’s a lot to be said for that position, but I’m not going to outline it here. The a priori nature of mathematics is not meant to answer the question, only to indicate why it might be that historical trends in mathematics would not suggest any particular answer.

So, really, the answer is, “There is no set answer, but there are apparently several different ones that satisfy different metaphysical views (which, themselves, are difficult to choose between).”

As a personal matter, I lean towards formalism, and consider mathematics as “the language of certainty”.

Mathochist, I’m not sure what you mean by “19th century mathematics”. Several mathematicians have also been philosophers (Russell is a prime example, as is Frege), and in fact, some philosophers even began in mathematics and later became philosophers (Husserl, for example). More complex mathematics do not make the philosophy any more difficult; indeed, given its deductive nature, once the primary elements are accounted for the rest more or less follows. Sometimes new mathematical methods spawn new debates, like the issue of infinitesimals, but even then these were often argued prior to their mathematical use (Zeno’s paradox of motion, for instance, is a good example). While I would agree that you don’t necessarily need to adopt any particular view to perform mathematics, I do not believe that this demands that people haven’t answered the question for themselves. Most famous mathematicians have, in fact, supported some view or another (Godel, Russel, Frege, Hilbert, etc.). It is not necessary to know how an automobile works in order to drive it, but that does not suggest to me that people aren’t commited to various positions on why it works (of course, neither does it compel anyone to care, but that is a different matter).

All of these wrote philosophy in the early 20th century to describe the mathematics being done at the end of the 19th.

But there are whole new fields that have come into their own since the Platonism/Formalism days. Categories and topoi have revolutionized mathematical logic, and Structuralism (which is practically built for them) incorporates these nicely. Modern abstract algebra, algebraic geometry, automorphic forms… actually, I think really all of these new vantages come back to categories and topoi if you consider why they start to outgrow Platonism/Formalism.

I don’t mean to say that it makes the philosophy more difficult, but if you’re trying to give a coherent philosophy of all mathematics while yourself understanding no more than high-school curriculum with a bit of calculus and first-order predicate logic thrown in you’re lost before you begin. Mostly, people who bother to learn this stuff are more concerned with doing mathematics than writing philosophy about it, so the question languishes.

That’s a little… understated. “What are numbers,” for instance, seems a little more broad than the 19th century, even if certain mathematical philosophers returned to such a question again.

What I fail to see is how platonism or formalism has been outgrown by the branches of mathematics you describe. What, specifically, do they fail to address?

The calculus really doesn’t add a lot to the debate, though, because so much of it really came from numbers and geometric objects themselves. The introduction of new objects or new relationships, as far as I can tell, does not instantly demand someone reconsider their stance on mathematical objects. I don’t believe most reconsidered realism after the discovery of the neutron, for example.

If you’d take the time to elaborate on structuralism as it pertains to the discussion, I for one would really appreciate it.

The problem is that, as far as the OP goes, Structuralism doesn’t really answer the question.

The best point of departure (IMHO) is the distinction between Zermelo-Fraenkel and Von Neumann numerals. For those who don’t know, the successor function for Z-F is

S(n) = n\union{n}

while for VN it’s

S(n) = {n}

So, in Z-F

0={}, 1={{}}, 2={{},{{}}}

while in VN

0={}, 1={{}}, 2={{{}}}

Now, in Z-F, 2 contains 0, but in VN, 2 doesn’t contain 0. From a straight Platonist standpoint this is a huge problem. Which is the “real” 2? Platonism says that 2 “exists”, and so it should have specific properties. Related is the “Caesar problem”, (which was considered by Frege, among others): The statements “2=Julius Caesar” is a perfectly well-formed statement in Platonism, but there is no way of sensibly determining its truth value, which must exist a priori.

Formalism falters on the universality of mathematics. This is a hazier point to make informally, but if you consider a mathematical statement to be just a bunch of inductions from formal axioms along certain well-defined formal pathways, why is it that everyone agrees on it? Why is it that (to a large extent, though not universally) a certain collection of axioms is the “right” one? After pondering this point, one is drawn to the feeling that something exists before a given thinker considers it. Even Formalists can’t deny that the Peano axioms describe a unique self-consistent system with “unreasonable effectiveness” in describing the physical world. Further, as we move into things like abstract algebra, Formalism is amazingly inept at talking about proofs of the form “for all groups G, G has property P”.

Structuralism is, in a sense, the outgrowth of Hilbert’s observation that the theorems of Euclidean geometry should hold true if everywhere one sees “point, line, plane”, one replaces it with “table, chair, beer mug” (“there exists through given pair of tables a unique chair”, etc). The idea is that, to go back to numbers, there is a unique structure of the natural numbers, which has slots for “0”, “1”, “2”, and so on, and a slot for the successor function. Both Z-F and VN numerals are structure-equivalent instantiations of this one structure, as would be any countably infinite collection of objects with an appropriately defined successor function. The only appropriate questions to ask of the natural numbers are those defined in terms of the structure. The structure of N, as outlined in the Peano axioms makes no reference whatsoever to set-theoretic containment, so though both VN and Z-F are set-theoretic instantiations, it makes no sense to even ask whether 2 contains 0, since that goes outside the structure.

Structuralism deals with things like group theory by considering the existance of the group structure. This is actually rather similar to Lawvere’s idea of Th(Grp): a category which encodes group theory, so that Grp (the category of groups) is the class of functors from Th(Grp) to Set. This structure is not unique, there being many structure-inequivalent instantiations of it (nonisomorphic groups). However, it does reduce statements about all groups to statements about the one object: Th(Grp). Group theory is the working out of properties of this one object.

The problem is that all we’ve done is change the venue. Structuralism is not monolithic: there are various interpretations. The two major schisms are between realism and nonrealism in ontology and between realism and nonrealism in truth value. Essential to the OP is the question of realism in ontology. That is: does the structure of the natural numbers itself exist a priori to be discovered, or was it invented as a method of describing and unifying its diverse instantiations? As subtler points: Does a given structure automatically have an instantiation by virtue of being self-consistent? Does a structure exist in and of itself, or only so far as it has instantiations?

There should be SDMB award for such elegant, concise answers.

Certainly the notation hasn’t been discovered. Sure, the notation is only an encoding of a discovery - but it is so useful that math would be nowhere without it.

Unfortunately I cannot mount any particular defense of platonism, since I don’t hold any realism to be the case in the first place, though having debated with enough platonists to sink a ship I have a good idea of what questions they’d ask are. But the “Julius Caesar” issue you raise in response to platonism is simply another form of the “why do we agree” problem you raise in response to formalism.

To which large extent? You just presented two mutually exclusive (in terms of their metaphysical interpretation) definitions of a successor function. Which do we all agree on? If we agree, why did you raise it as a problem for platonists? If we don’t agree, why are you telling the formalist we do?

As a formalist, I ask, “When you do select among various alternatives, what is your criteria?” As a platonist, I’d ask, “Why do you call both of those successors ‘2’?”

Your comment on structuralism reminds me of the coherence theory of truth. It still leaves the platonist wondering, “Do mathematical structures exist?” You’ve shifted the scope of the question, but not made it disappear. The Russell quote I gave in the previous post is something that has always struck me. Yes, numbers are, in a certain sense, formal; but by that I simply agree to, “Numbers, in a certain sense, can be formalized.” Were I a platonist, I would not suggest that any particular representation of ‘2’ somehow captured it once and for all; indeed, that would more or less fall right into immanent realism. A particular triangle drawn for the purpose of a proof about all triangles in plane geometry is not meant to limit all triangles to acute angles, obtuse angles, etc. As we investigate mathematical objects, we might be surprised to discover new objects and properties, which may in fact cause us to reappraise our previous notions in terms of which objects are fundamental, but not necessarily reappraise our answer to the question, “How do mathematical objects exist?”

But the point, I think, is that while the different sets of axioms are equivalent in that they give the same results. In Set Theory, anything that’s true under the Zermelo-Frankel axioms is true under Von Neumann-Bernays-Godel axioms, and vice versa, even though the axioms themselves are different. But what Mathocist is asking (I think – I’m sure he’ll correct me if I’m wrong) is if the formalists are right and the axioms are just an invention with no underlying truth (*), then why is it that when someone comes along with a different set of axioms which give different results, we all immediately recognize these as incorrect. Why is there agreement that the axioms of set theory must be equivalent to the Zermelo-Frankel axioms? (Or likewise, why is there agreement that the Peano axioms or some equivalent set of axioms correctly describe the natural numbers, etc.)

**(*)**I don’t really know much about the Formalism vs. Platonism debate, but I’ve inferred from the discussion in this thread that this is the position of Formalism. If I’ve misunderstood, I’m sure you’ll let me know.

Not so fast there, chief. There are a number of alternate versions of axiomatic geometry, the most famous being the three that arise from various alterations of Postulate V. There are less well-known ones stemming from alterations of other assumptions.

A Structuralist would say that the structure defined by the Peano axioms is such that any two instantiations are structure-equivalent (“the same thing”. A uniqueness result), and that Z-F or VN numerals are an example of an instantiation (an existance argument), and so there is an essentially uniquely defined N. The fact that it reflects the properties of objects in the physical world we first encounter through counting is an example of the “unreasonable effectiveness” that Formalism completely fails to account for.
In order to recap, for those like tim who need a refresher:
Platonism (in mathematics) says that all mathematical concepts (as well as all concepts or “forms” in general) have an independant existance in what’s referred to as “Platonic Heaven”. There the concept of 2 exists and all collections of two objects reflect some aspect of this “form”.

Formalism says that none of mathematics has any real meaning, and it’s all a process of taking some collection of axioms and another collection of rules of inference and putting together established results (starting with the axioms) via those rules to form new ones. In theory, any result could be reduced to a very long chain of symbolic manipulations.

Anything anyone else thinks should be mentioned about these two old warhorses?