Mathematical Realism and 0=1

Question

Mathematical Realism is the notion that mathematical truth exists, and is not subjective or merely a mental construction.

Inspired by Noah Schweber’s recent post on Math Stack Exchange: https://math.stackexchange.com/questions/4942650/functional-completeness-over-a-structure, I wonder whether 0=1 is evidence that mathematics is real.

That is, since an entire class of functions on seemingly arbitrary objects exist that can exactly model negation for their respective theories, is it not the case that things like 0=1 are just set-in-stone mathematical phenomena about which there are necessary truths? Take for example the functional completeness mentioned in Noah’s question.

What are some Anti-Realist and/or Constructivist approaches to such phenomena?

Edit:

All I’m saying is that humans can’t just decide to ignore that falsity as our formal systems can represent it is unavoidable in arithmetic contexts, even without assuming anything to represent falsity at the logical level.

Answer 1

I understand what you're getting at with the 0 = 1 thing. You're saying that we can represent negation of a proposition S as S → ⊥, where ⊥ is set to 0 = 1. And the empty set is the set with no elements, which per Frege is supposedly because it contains all self-contradictory elements, except "of course nothing self-contradictory actually exists." But so the relationship between zero, the empty set, contradictions, falsity, etc. is there, it's intuitive, and what is it an intuition of?

Here's a contentious proposal which I haven't found a citation for, yet (give me time, I'm sure it's out there somewhere...):

• The real overarching debate, in the philosophy of mathematics, is not between realism and non-realism, but between ante rem and in re realism.

The one is usually known as "Platonic," the other "Aristotelian." But the point is that appeals to intuitions, fictions, modal dimensions, a priori games/symbolism, etc. are not only mostly interchangeable, in practice, with each other, but they are not even legitimately describable as "non-real," ultimately. For if we really intuit that or really narrate that or really play the game such that, or whatever, then there are intuitions, narrations, games, and the like. But they are in re things, not floating in some other world. At "worst," in the awesome light of multiversal standpoints, we can say that it is some confluence of abstract free will and pure conceptions of things that is at play here, but we still need not absolutely have our will occupying a place in another world, do we?

Still, we would like to know whether it is merely "conventional" or more like "naturally isomorphic," that 0 = 1 and negation are related as they are. Realism isn't just about "things," but about objectivity, at least to a decent extent. General murkiness aside, let's say that even if the exact details of this case are "conventional," they reflect a natural "theme" of using two different items flagging an equation sign as a primitive sentential representation of falsity. Like, 0 = 1 itself is the first false non-negative sentence possible in basic arithmetic (the first negative one would be 0 ≠ 0, I suppose).

Then that's the next layer of the dialectic: granting that either way, some sort of realism is at hand, here, are more particular cases of this phenomenon trivial or not? For it is often a sign of a theory's maturity, that it is able to pass beyond fretting over whether it is a false or a true theory, to trying to evolve into a non-trivial notion of the way things are. But so 0 = 1 is a "trivially" false sentence; or, it's a "base case" of falsity (in base-case arithmetic, or arithmetic-as-a-base-case itself!). So it might support a trivial realism, but not a very substantive one, here. (Unless, of course, triviality is turned upon itself, as a matter of inquiry: for then it is not necessarily externally trivial that it is internally trivial that 0 = 1 has a "realistic" significance.)

Answer 2

As a preface, according to Dedekind-Peano, the existence of 0 is taken as an axiomatic truth and 1 is defined as the successor function. Equality of naturals becomes then a question of whether they have the same structure in regards to successor functions. This is m=n<-succ(m)=succ(n). The idea that a constructivist accepts any syntactic construction such as 0=1 is a bad characterization. Rather, constructivism makes proof a central element of ontological commitment. If you can construct it, then it exists. Realism doesn't have that criterion. So, constructivists are quite comfortable with the system PA too and wouldn't accept 0=1.

That being said, you ask:

is it not the case that things like 0=1 are just set-in-stone mathematical phenomena about which there are genuine truths?

It depends on what you mean by "set-in-stone mathematical truths". The debate between realists and those who reject realism is about the nature of those truths with the debate taking on the same overtones as various externalism-internalism debates. Are those truths internal to the mind or are they external to the mind, and if so, how does one account for them being "set-in-stone" or more in line with canonical terminology accept them as necessary truths (SEP). Thus, the difference between a realist and constructivist is in how to view the origin of the truths you refer to.

A realist will simply say that 0≠1 because the truth is external to the mind of the agent doing the arithmetic. What "external to the mind of the agent" means is then up to the realist to defend. This requires a realist philosophical position that explains how the truth becomes known to the mind. The famous and prototypical example of such a philosophical theory is Plato's Theory of Forms.

But what if you reject the Theory of Forms or other externalist arguments? How does one account for the necessity of mathematical truths such as 0≠1? Sometimes in the same way a Platonist does. A constructivist can use the logical axioms of PA to show that 0≠1 in the same way a realist does. The statement '2+2=4' is likewise true since succ(succ(0))+succ(succ(0))=succ(succ(succ(succ(0)))). But if arithmetic is grounded in logic (a truly historical event), how do we account for logic formulae?

One option is to accept a philosophically contentious notion that logical formulae are by-products of the human body and brain. In this view, the neurons of the brain perform calculations that form the basis of arithmetical and logical operations. That is to say, the mind is somehow the product of the body and the brain's function. Of course, this is squarely an argument that comes from the philosophy of mind, and there are various ways of arguing that: identity theory (IEP), functionalism (IEP), and the computational theory of mind (IEP) are three examples that serve as a basis to defend an internalist thesis in regards to the internal nature of the necessary mathematical and logical truths.

A specific philosophical theory comes from Philosophy in the Flesh (GB) by George Lakoff which puts forth a position he labels 'embodied realism'. In the theory, he makes the claim that neural computations construct the basic operations of the mind, for him and his ilk conceptual metaphors, these operations construct our language that is used to do logic and arithmetic. Thus, instead of the Theory of Forms, we have something akin to a "Theory of Psychological Conceptual Primitives".

So, of course Lakoff is one anti-realist in regards to the origins of formal semantics and formal systems. There are others, and what all anti-realists have in common is they reject the arguments of anti-psychologism.

Answer 3

There seem to be two layers, which I could be wrong about:

(1) Does an encompassing and interesting mathematical fact motivate classical realism in particular: "an entire class of functions on seemingly arbitrary objects exist that can exactly model negation for their respective theories...set-in-stone mathematical phenomena about which there are necessary truths"

(2) 0=1: I have no full idea what role 0=1 is playing here. Are you saying some mathematical real facts are contradictory , a la "really full-blooded mathematical platonism" like JC Beall? Or are you doing something else with that notation? I'm taking it to stand-in for a large mathematical fact with some abuse of notation, rather than as a real contradiction. It seems like in any case (2) is less important than (1), i.e. the meat of your question is, do large necessary truths point toward realism?

And assuming there's nothing particularly special about your encompassing mathematical fact (I don't see what's particularly special about Noah's example in regards to realism), then the usual arguments apply in face of an interesting mathematical fact:

• Nominalists typically don't want to overcommit to ontology (e.g. maybe I don't want to commit to uncountably infinite objects since no physical means can ever surveil it) or commmit to non non-spatiotemporal objects (like abstract mathematical objects typically area via realism). If there is a non-committing alternative to platonic realism they will choose it, like modal structuralism (it's possible such a position in a structure exists) or fictionalism (the object doesn't technically exist but we speak by what's convenient).

• Contstructivists don't want their logical symbols to denote timeless truth whereas classical mathematicians do. Classical mathematicians will say, look at this timeless truth! And the constructivist will reply, I only know what constructive truth is. Whence some important conjecture is proven, why speak of a further realm where there is an object corresponding in the right way to the proof, making it platonically real?

These are immediate, surface level responses and the conversation between sides would get more in-depth. I don't see what this mathematical fact does in particular in the debate of realism vs anti-realism and constructivism (and what are you getting at with 0=1 exactly?).

We see the necessity of math is not enough to declare realism. The necessity of math is not solely identified with its facts corresponding to abstract objects. We can get necessity via many means (by if-thenism, as-ifism, fictional games, different semantics and syntax, collective behavior, etc).