I think first that we need to clarify the concept of truth (or “objectivity” if you wish) of mathematical proofs, and distinguish such concept from the debate around mathematical ontological realism (which explicitly addresses the “existence” of “mathematical objects”); they are two separate and different issues.

If your opposition to mathematical realism refers to the latter, then I am OK with it, in a qualified manner (this would be a whole different discussion, which I am not addressing here). On the other hand, I feel the need to address the former (“objectivity of mathematical proof”) in more detail, as this is where I strongly dispute the claim of an identification between objectivity and acceptance/popularity.

Firstly, I think it important that we need to avoid confusing the issue of objectivity of formal systems (axioms etc.) from the objectivity of mathematical proof for a given formal system. They are two different things. A formal system, by the way, does not necessarily need to be a given set theory (some mathematicians have been exploring the alternative option of category theory, for example), but you always have to refer to one, in modern mathematics, either implicitly or explicitly. Very few mathematicians, if any, will state that that ZFC “is true”, but no serious mathematician will give you a mathematical proof without referring, explicitly or implicitly, to the underlying assumptions either.

A mathematical proof is not a question of “acceptance” by mathematicians. That there is a infinite number of primes is true not because mathematicians say so, but because there are incontrovertible proofs of it (within a given system). This is a statement of fact, not an opinion subject to the vagaries of a supposed majority of experts. You will never find an article by a mathematician arguing for the statement that there is a finite number of primes (within ZFC, for example).

Within a given axiomatic system, the truth-value of a mathematical statement that is reachable by it is not a question of a poll of opinions: it is objectively true or not, and once the underlying axioms are accepted, a mathematical proof is intrinsically correct or not.

This does not imply of course that one set of axioms is per se objectively true or not. This is al altogether different question. Please refer to my original comment regarding some of the motivations behind one choice of axioms over another. This is where an issue of choice (subjectivity) arises. But, even if all mathematicians, with no exception, agreed for example that ZFC is “the system” to use, it would still not be “proven” nor considered “objective” – and no mathematician would ever define ZFC as objectively true – I think that we should not confuse popularity or acceptance with objectivity, both in the case of formal systems as well as in the case of generation of mathematical proofs.

I agree that without minds (brains or something functionally equivalent) there would not be any mathematics – absolutely. In this case, there would also be no science either, mind you, and the whole thing would become purely academic. BUT, if there are “minds” they will all have to ultimately agree, within a given formal system, about the correctness or not of a given mathematical proof.

This would be true even of the formal system itself is inconsistent! If you could prove that a particular statement is both true and not true within such system, this still would not remove any objectivity to the correctness of both proofs. It would only confirm the inconsistency of such system. Similarly, the fact that a formal system can’t prove its own consistency (Godel) nor that it is not necessarily complete (Godel again) does not have any relevance to this.

Once a set of rules is adopted as a conscious choice by an intelligent entity, the position within such system of a given theorem (assuming it is reachable by the chosen formal system) is objective, independent of such entity. If you transmitted your axioms and rules to a completely different alien civilization, they would agree with you on whether a specific derivation of a proof is true or not.

For example, once you define the concept of number, of integer etc. and of prime number in the way defined by our most accepted current formal systems, there is no way you can demonstrate that the number 11 is not a prime number. This is tautological in its truth-value. If you want to make an opposite claim, you must choose a different formal system. You might even manage to design a formal system in such a way that the number 11 is not a prime, which would then be there an objectively true statement in the same way as 11 is a prime is an objectively true statement in my system.

It is also important to highlight what “true” in mathematics really means – it means compatible, consistent with the derivation rules and axioms stated in your formal system.

I can give you many “true” mathematical statements that may appear paradoxical if compared with the physical world (for example, the famous Banach and Tarski result, generated from the application of the axiom of choice). As I said before it is like a game; in the case of checkers, for example, you know that you will never find objects both on black and white squares – this is not something subject to opinion polls, once the rules are defined. In the example of checkers, your formal system would be: the topology of the board, the number of type of pieces, the initial configuration and the possible moves, and the condition victory. Once this is defined, any given “theorem” (configuration of pieces on the board) would either be “true” (configuration of pieces allowed) or “false” (configuration of pieces not congruent with the rules).

If on the other hand by “truth” in mathematics you refer to correspondence with the physical world, then you would adopt an extremely limited and partial view of modern mathematics, a view that would cut off huge swathes of modern mathematics indeed. Even with Euclid at the beginning of mathematical thought we had idealizations (such as the zero-dimensional point) that do not correspond with anything real in the physical world.

I think that it is also of fundamental importance to highlight that, while at first superficial glance there does not appear any direct link between the most complex and convoluted theorems of mathematics and the underlying axioms and rules of the formal system used, in reality such links are always defined and present, even if most often not explicitly stated or referenced.

This is not because there is no such link, it is simply because mathematics is built, for its very nature, on blocks over blocks over blocks of elements, and for readability and convenience you do not want every time to go all the way back to the most basic atomic elements. More frequently that not, you refer to the highest levels of the conceptual hierarchy, rather than going back to the features say of the field of the complex numbers. But this does not remove the basic fact that the proof of any proposition must be theoretically traceable back to these axioms, whatever they are.

Let me make an example, which is a big over-simplification but still gives you the flavour of the problem: the mathematical apparatus of calculus is often used as one of the tools of higher mathematics – it is assumed “true”, not re-discovered or re-demonstrated. Calculus itself is based on more primitive concepts of real analysis such as continuity, infinity etc.. and such concepts can be directly related to more primitive items such as the basic features of the field of the reals and ultimately to the axioms of your formals system such as ZFC.

There is also another element to this debate, which is the historical development of human mathematics, which a separate issue but still relevant to this discussion; the reason why mathematics developed in a particular way rather than another is due to many causes, some of them even due to pure historical accident and in many cases simply because of very practical administrative and commercial requirements (this is how written language started too, after all). And mathematics surely did not develop historically in a tidy, ordered, coordinated way starting from a robust set of axioms all the way to the latest results. Moreover, mathematical physics certainly does not put much focus on mathematical rigour – but where shortcuts are taken (like in the case of re-normalization in QFT, or in the usage of the Dirac Delta “function”) in almost all such cases the practitioners doing this are aware of the consequences of such actions and manage them appropriately. But this does not mean that the logical and conceptual structure of modern mathematics does not depend on the choice of a set of initial axioms or definitions – a formal system if you wish.

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]]>First, I have nothing to argue against as far as the content of your first paragraph. I will make a few comments nevertheless.

I am aware of the axioms of set theory, and that they are use to build up the whole of mathematics, ideally. I say it is “ideally” because I do not think that anyone has actually accomplish this. And, there are many mathematicians that work outside the confines of set theory, but their work is still logically produce.

I still think it is appropriate to claim that mathematical objectivity (any type actually) resides with the acceptance of the mathematical community. This does not mean that the axioms of set theory are not objective; the objectivity of it just resides in human mind’s unless you are talking about applied mathematics, which may or may not apply to the real world, but that is what it is tested against as you talk about in you comment. Without minds (brains or something functionally equivalent) there would not be any mathematics. This assumes that mathematical realism is false as I argued for in this post. A not exact thing happens in scientific research. The research is reviewed by qualified peers, and then if it is found to be acceptable it becomes part of scientific knowledge (i.e. objective). The main difference is that with science you are looking at physical objects and operations on them. Maybe, I should only claim the recognition of objectivity, but with mathematics were would it reside; I still do not accept mathematical realism.

My argument against mathematical realism was against the ontological status of mathematical objects or truth. There is just not a trace of evidence besides some mathematicians desires, which from my perspective does not accomplish any objectivity.

Your presentation of structuralism is surely interesting, and from are previous conversations it does not surprise me that you would bring it up in your comment. I am still thinking and feeling my way about this position, so I will not comment further about it here. I think you reviewed a book (on goodreads) that discussed one version of structuralism. I cannot remember the title. I want to reread this review because my first reading involved some skimming, so maybe you could post the title of the book here or in a goodreads message. I probably cannot afford to by a copy. Do you think a public library would have it?

Thanks again for your comments. The are deeply appreciate them, and I am grateful that you chose to post them.

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]]>I never thought that it was a satire. I am not even sure how one would write one.

That being said, I try to inject humor in my writing. After all, a lot of philosophical writing is dry. I want to entertain my readers minds with my thoughts that I put to paper,* and part of my thoughts focus on the humorous.

As for the eating of the fruit today, it would be better if people thought of the story as a fairy tale, instead of accepting the horrible doctrine of original sin, which when added to the Jesus fairytale damns people to an eternal life of burning in a trash heap for those that will not accept Jesus as their savior even if they were born and died before Jesus appeared on the scene or have never heard of him. (Jeez, this is one of the longest sentences I have ever written)

Thanks again for your comment

* Actually, I do not put anything to paper (it just sounds better); I type everything on the computer on the my WordPress page/site, except for doing a print revision, which then gets entered in my blog post my laptop or my phone on occasion when I am out and have something to write that I thought of and do not want to forget it.

]]>There are a couple of points, in relation to the issue of “mathematical realism”, that I would like to highlight, in particular when you referred to the concept of mathematical “objectivity”, stating that:

“It becomes objective, however, when mathematical proofs have been accepted by the mathematical community. In other words you could say it becomes part of the public domain, and that is where mathematics’ objectivity resides.”

I would like to qualify this view of “mathematical objectivity”.

Modern mathematics is essentially based on axiomatic foundations – once you start with a series of axioms, a language and a set of rules (for example, ZFC is based on an example of first order predicate logic), then you have objectively and unequivocally defined, once and for all, what criteria determine the derivation of “theorems”. That is completely independent of “social” factors such as the acceptance by mathematicians.

From this perspective, I think that Hilbert was not wrong when he claimed that mathematics is a “formula game”; it is a game like chess where, once you have defined the rules and the context, what are the possible outcomes is unequivocally defined.

There is still “subjectivity”, of course, but it resides only in the choice of axioms and language/rules. For example, most mathematicians accept ZFC as the preferred axiomatic system, but this is a question of choice depending on many factors, including flexibility, convenience and usefulness.

Even this choice however is not as arbitrary as it may seem at first glance, though. In fact, regarding the latter criteria (usefulness) it is important not to assume that there is a mono-directional relationship between mathematics and the physical sciences. The selection of the preferred axiomatic system, and the actual evolution of mathematical thought, have historically depended not just on pen-and-paper pure mathematical analysis by mathematicians, but it has been significantly influenced by the very down-to-Earth empirical results and demands coming from the experimental sciences.

There is therefore significant implicit empirical content and empirical legitimacy in much of mathematical thought – whose “objectivity” and epistemological value can therefore be closely related to that characteristic of the physical sciences. Mathematics can sometimes “drive” the physical sciences (like in the cases of Dirac’s equation, for example), but the opposite is true too. Among all “games” that you may want to play within mathematics, only few such games (choice if axiomatic system) will give you results that are empirically useful.

Regarding the issue of mathematical realism, it must be said that the naive mathematical-realist view that mathematical objects “exist” in some “conceptual” world beyond the spacetime of the natural world belongs to just a minority view.

More modern approaches are often based on some form of “structuralism” – which, contrarily to naive mathematical realism, is more an epistemologically realistic view rather than an ontologically realistic view, at least with regards to mathematical objects.

Structuralism shifts the focus from a naive view of mathematical objects to the underlying patterns and structures that clearly characterise physical reality (science is nothing but discovering such patterns and structures, after all), patterns that mathematics tries to represent.

Structuralism tends to highlight, and most importantly clarify, the confusion between the four different levels of discourse that all too frequently arise, when discussing mathematical realism and philosophy of mathematics in general:

– human mathematics (mathematics historically developed so far by the human species, influenced also by factors such as historical accident and the idiosyncrasies of the human perceptual and mental apparatus)

– “Mathematics” in general, as a general language that can represents the regularities and structures of physical reality

– Structures and Patterns (laws, if you wish) that characterise physical reality

– Physical reality itself

Structuralism itself is far from being a monolithic approach: there are many different flavors, each positing a different type of relationship between each of the 4 levels mentioned above: starting from the ante rem structuralism (which has a ontology in some aspects compatible with Platonism, and where “structures” are considered ontologically primary), all the way to various forms of in re structuralism (“in the thing”, which can be compared to Aristotele-an realism, where structures are exist only within “concrete system” implementing them).

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]]>I am pleasantly surprise that you or anyone else would agree that we do not have minds.

For me there is a difference in the non-existence of the mind and the non-existence of the soul.

The soul is based on the concept of god for which I see no good reason for believing in, while at least the mind is based on the brain for which we have plenty of evidence for..

Oops. I cannot be a deep thinker than anyone else, since we do not think because this is another folk psychology concept, which needs to be rejected according to eliminative materialism. And, remember thinking is one of those things the mind is supposed to do. So, no mind, no thinking either. This just illustrates the massive task it would be to eliminate folk psychology from our usage of it.

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