# Are theorems of math theorems even before they are proven?

I considered asking this in the math SE, but I decided this was a better option. I got into an argument with a math professor who claimed that Fermat's Last Theorem was a theorem only after it was proven in the 1990s. He didn't claim it had not been a true statement before the proof, but merely that it had not been a theorem. I claimed that all theorems of math are in fact theorems timelessly, because a theorem means "provable", not "has been proven". So, is this correct? Are theorems of math theorems even before they are proven?

• It sounds more or less correct, assuming the timelessness of mathematical truth, since all the axioms and rules of inference, being such, would timelessly, necessarily imply the conclusion, here. But to an intuitionist, it wouldn't quite sound correct. A fictionalist or formalist might say that a proposition is a theorem only when the story or game of mathematics has reached the point where the proposition has actually been proven, too. Mar 12 at 2:45
• If you both agree the statement was true before it was proven, and the disagreement is just over whether it was a "theorem", this sounds like purely a question of terminology, not any kind of substantive philosophical disagreement. Mar 12 at 3:08
• Theorem does have a distinctive feature which only occurs after it's been proven, that is it may acquire a name usually honoring its most important originator(s). This addition does have some substantive importance for some professionals as it embodies certain philosophical doctrines... Mar 12 at 3:34
• I’m voting to close this question because belongs on Math SE Mar 12 at 4:55
• The OP definition is the one used in mathematical logic and the more abstract side of mathematics. As for the use in the vernacular, opinions differ. Wikipedia ("a theorem is a non-self-evident statement that has been proven to be true") and MathWorld ("A theorem is a statement that can be demonstrated to be true") that usually mirror each other disagree on this one. Hypnosifl is right, this is a matter of terminological choice, there is no "correct" choice for what the word "theorem" should denote. Mar 12 at 7:16

In most mathematical usage no, and this is purely a linguistic question. Theorems are true before they are proven, but not yet theorems. The word "theorem" usually means not just a provable proposition, but a proposition that has already been proven and given a name, which often names the discoverer, e.g. "Haboush's theorem." You wouldn't say that Haboush's theorem was Haboush's theorem before it was discovered, for the same reason you wouldn't say Canada was Canada before it was colonized.

wikipedia says: "In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true"

wiktionary says: "A mathematical statement of some importance that has been proven to be true."

wolfram mathworld says: "A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments." This definition seems to contradict the wiktionary/wikipedia ones, but I believe the proper reading of "can be demonstrated to be true" is the pragmatic one, that "a demonstration is already available to present," which yields no contradiction.

Britannica says: "Theorem, in mathematics and logic, a proposition or statement that is demonstrated."

See also the wikipedia list of conjectures. Some theorems are still called conjectures by convention even after their proof. (And some important theorems are called lemmas, again just out of tradition.)

Edit: However, sometimes the other sense is used, in which a "theorem" refers to anything provable, even if not proved. This is usually in reference to a particular formal system, e.g. "the theorems of first order logic" refers to every statement provable in first order logic. It doesn't tend to be used in the context of named theorems in the rest of mathematics. An example of that is here: "Gödel's completeness theorem established equivalence between valid formulas of first-order predicate calculus and formal theorems of first-order predicate calculus."

• +1 for mentioning it is a purely linguistic question. Mar 12 at 10:36

In math, facts just thought/observed to be true (basically in a model) but not yet proven are usually called conjectures, although there may be occasional departures from this (terminological) convention. Wikipedia actually mentions FLT as example:

In mathematics, a conjecture is a conclusion or a proposition which is suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

FLT was named so because Fermat claimed (at one point) he actually had proved it, even though he later didn't mention that again (perhaps because he found a flaw in his proof), so there's definitely a sociological angle to that name. As Wikipedia recounts:

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem.

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the margin note became known over time as Fermat’s Last Theorem, as it was the last of Fermat's asserted theorems to remain unproved.

It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4, as described in the section Proofs for specific exponents. While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis, he never posed the general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten suggests that while the absence of a proof is insignificant, the lack of challenges [i.e. not posing it as a problem to others in correspondence] means Fermat realised he did not have a proof.

Due to incompleteness results (such as Gödel's) and independence results (e.g. CH from ZFC), the status of [unproven] conjectures e.g. Goldbach's or "extra" axioms like CH as truths in some (other) sense is more debatable, but no mathematician would argue they are theorems in any formal sense. (Aside: the independence of CH from ZFC was conjectured by Gödel, but proven by Cohen.)

On the other hand, the process of discovery of such conjectures, before their proof, has been linked to a scientific one, e.g. as:

a mathematical version of the well-known philosophical problem of induction. A large part of mathematical research consists in spotting patterns, making conjectures, guessing general statements after examining a few specific instances, and so on. In other words, mathematicians practise induction in the scientific as well as mathematical sense.

It is more debated what kind of ontological status such conjectures [or axioms] have (besides the fact that they are not theorems). Gödel himself, for instance, argued that:

“There might exist axioms so abundant in their verifiable consequences, shedding so much light on a whole field, yielding such powerful methods for solving problems […] that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory”. (Gödel 1947, p. 477).

This has been ascribed to some kind of (utility-based, at least) form of Platonism; at least SEP does that. One could also argue there's a form of social constructivism in there as well: some axioms or conjectures are interesting because people agree (on some level) that they are. However, mathematicians usually agree on something like that because of interconnections with other results, which in some Platonic sense can't simply be a social construction.

The SEP page on the non-deductive aspects of mathematics delves into these matters, e.g.:

As James Franklin puts it:

Mathematics cannot consist just of conjectures, refutations and proofs. Anyone can generate conjectures, but which ones are worth investigating? … Which might be capable of proof by a method in the mathematician’s repertoire? … Which are unlikely to yield answer until after the next review of tenure? [...]

It is also possible to find passages along similar lines in the work of Pólya, who was a major influence on Lakatos:

Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science, but mathematics in the making appears an experimental, inductive science. (Pólya 1945, vii)

Getting to more concrete examples, such as Goldbach's conjecture, one can find interesting passages, e.g.

What makes this situation especially interesting is that mathematicians have long been confident in the truth of GC. Hardy & Littlewood asserted, back in 1922, that “there is no reasonable doubt that the theorem is correct,” and Echeverria, in a recent survey article, writes that “the certainty of mathematicians about the truth of GC is complete” (Echeverria 1996, 42). Moreover this confidence in the truth of GC is typically linked explicitly to the inductive evidence: for instance, G.H. Hardy described the numerical evidence supporting the truth of GC as “overwhelming.” Thus it seems reasonable to conclude that the grounds for mathematicians’ belief in GC is the enumerative inductive evidence.

(The passage above also gives another, albeit uncommon, example of the word "theorem" being [improperly] used to denote a conjecture.)

It would be interesting to detail what status various "standard" philosophies of mathematics assign to conjectures, but I'm not sufficiently versed in that (and the main SEP page on philosophy of mathematics kinda avoids the topic).

Axioms began as 'self evidently true statements', for Euclid. Needless to say that doesn't cut it now. Have a watch of this quick introduction to the philosophy of mathematics, for a quick portrait of how far things have changed.

It used to be thought that numbers are 'just out there', and mathematical Platonists still think that. But mathematics is recognised now largely by philosophers, as operating within definitions, and those shift, develop, change. For instance due to metamathematics like group theory, and Godel's theorems. So recognised as a human-made structure of abstractions, how can we say 'always true'? We can only say: were always implications of a given set of assumptions/definitions, related to a certain level of understanding. And there will be more, just as science deepens, and integrates. There is no ultimate grounding or foundation, to judge timeless truth.

I would argue we only expand the coherentism basis of mathematics, that this has been the nature of development. To fit fields together, and find unifying abstractions like set theory which can encompass many areas as special cases, and provide mappings between them.

In philosophy, the Private Language Argument, attacks the idea of a-priori meaning, accessible as an individual privately through introspection. Without that, how can there be any 'always true'? There can only be 'transcendental truth' in so far as there is 'transcendental context', and things like the possibility of substrate-independent artificial general intelligences, challenge our idea of how universal a context can be assumed, as do the possibilities for alien intelligences. Human intelligence occurs in an intersubjective space, where we build a network of interactions that allow us to know we share experiences from scribbles on paper, or noises from our faces, and accumulate & inherit knowledge over many generations. By which our meanings transcend individual lives, but cannot go beyond what we experience and imagine together, until we can reach new contexts, definitions, methodologies & make them shareable. Our truths are limited by context, definitions, and the hidden constraints of shared modes of life required for our low-bandwidth communication. Consider, what an 11 dimensional being living in a string-theoretic universe would think of our 'truths'.