In this blog post I will explore what realism entails, whether or not it is a valid claim in any domain. At the moment I have in mind four types of realism: Platonic, mathematical, scientific, and artistic. Platonic realism concerns where Plato’s forms or ideas resided. The forms were considered the true reality, where those things in the everyday world were only considered faint copies. Mathematical realism is more or less a form of Platonic realism. The main difference is that only mathematical truths reside there. Scientific realism claims that what science has shown to exists does indeed exists, such as atoms. Finally, artistic realism concerns artwork which comes as close to the everyday world as the artist can make it, showing as much detail as possible. I will take each of these in turn and speak on whether or not I see them as holding some aspect of truth to them.

I will start with Platonic realism. This form of realism asks us to see the everyday world around us as an imperfect reflection of the true forms that exist in some type of heaven (i.e. a non-physical other world). These forms are also sometimes referred to as ideas. I guess this might be because they are the ideals as in being perfect specimens. This is why nothing in our everyday world is perfect according to this type of realism.

So, why did Plato take this slant. One reason is try as you will you cannot draw a line because it has no width, only length. The same can be said for any other geometrical figure. There is no way to draw the perfect anything, let alone a geometrical figure. So, the idea of say a circle needs to have a perfect form. And, all these forms reside in the *wherever*. I mention geometry here because it was said that “let no one ignorant of geometry enter here” was placed above the entrance to his academy—at least this what is usually said.

But, these are not the only things that reside the Platonic realm. There are also things like virtues. These are concepts such as justice, happiness, and honesty. The highest of all of these is “the good,” and this not geometry may have been the ultimate reason for Plato to introduce us to his perfect world of forms. We may not be able to determine the virtues’ exact nature in the world of forms because Socrates was famous in Plato’s dialogues for asking about these things and never being able pin down their perfect nature. There always seemed to be a hitch to whatever definition Socrates and his interlocutors arrived at. Even if they could have been successful, whatever can be observed in our everyday world is a pale comparison to what resides in the world of forms, so no one would be able to exhibit any type of perfect virtue. But, you can rest assured that they exist.

In addition, the perfect instances of everyday objects also reside in this Platonic world of forms. You could find the perfect table, chair, bed, or house there, encompassing the ideal of each object. These objects we see in our everyday world pale in comparison once again with those existing in the world of forms. You would also need to include the living world, so you will also find the ideal cat, mouse, elephant, or any other creature there.

So, how did Plato know that there was a realm of true forms where reality really resided? He did not, but he did make up a story that I will not relate accept to give you the title, and where it can be found. I have become bored with it after having heard it related in so many books. The name of the story is the “Allegory of the Cave” in his *Republic*. It is a big book, so here is a link to just the story – https://web.stanford.edu/class/ihum40/cave.pdf. Regardless, the story is just—a story and nothing more; it does not prove anything. Although I do not know if it was intended to be a proof or not, it could be seen as a way of illustrating a better world than our own.

One attraction to the theory of forms involves the question of universals. In other words how do we form general concepts. According to this theory you find them in the Platonic realm—plain and simple. Aristotle did not see universals as presenting much of a problem, so he argued against the need for the Platonic solution. My own sense is all we are doing with general concepts is giving a name for a class of individuals (even abstract ones). All cats makes up the class of cats, all houses makes up the class of houses, all triangles make up the class of triangles, and so on and so forth. In the late middle ages William of Ockham supported this general view on universals, although maybe not in this form (the concept of classes was not formed back then), and it became known as nominalism. And, its opposite is realism of which Plato’s theory is just one. Another oppositional view to both of these is idealism—all reality is in the mind. But, I am not going to criticize idealism in this post (except – see below); although, there is plenty to criticize. I think the “class” thing is probably the best way to deal with universals, but again I am not discussing this here.

Here are my reasons for not believing in the theory of forms with true reality dwelling elsewhere than in our everyday world. My main issue with this theory is where is this reality supposed to be. Obviously, not in Plato’s cave. Even if forms did reside in another plane than the everyday world, how could we gain access to them. That to me is a big problem. What the universe is made of and how it works is the domain of science. And, science, at the very least in practice, takes a naturalist view of the universe. Science, almost by definition, cannot investigate the Platonic realm of true forms, hence why we have no access to it. Another thing is, is there really a perfect thing anything?* What about geometric figures? Should not there be a perfect triangle? No. A triangle is a definition, it has no reality in itself; it only shows what is necessary for something to be called a triangle.

Well, I think I have more or less dealt with Platonic realism. So, what about mathematical realism. People, mathematicians mainly, who believe in mathematical realism believe that there is a realm where all mathematical truths reside. Some also believe that numbers themselves actually reside there. But, I have heard it described both ways. This realism is very similar to Platonic realism, except the mathematical realist is only concerned with mathematical truths or objects, not with regular objects (e.g. houses, cats, mice, etc.) or virtues (e.g. justice, honesty, courage, etc.); although, mathematicians find great virtue in proof, as well they should.

Plato was a mathematical realist, as were Gottlob Frege, Kurt Gödel, and G. H. Hardy. Frege was the first to attempt to put mathematics into a logical form. He was not successful. First, Bertrand Russell pointed out a contradiction within his system, which Russell himself attempted to correct, but Gödel put a nail in the coffin of logicalism with his theorems (consistency and incompleteness). However, Gödel still was a realist when it came to mathematics at least. Hardy was a superstar of number theory. He consider even the hint of applied mathematics as a sin against real mathematics. He felt his research into pure numbers had no practical applications (you can argue that he was mistaken). So, where else would he find mathematics other than in a realm of purity.

Mathematical realism often comes down to whether mathematics is discovered or invented. For the discovery camp they feel that there must be something out there to be discovered, hence their liking for another realm where there is waiting to be discovered mathematical truths or the properties of numbers. Under discovery if one is to go about discovering mathematical truths, like a scientist discovering atoms (more on atoms below) one must find them somewhere. Might the mathematical realist suffer from a touch of science envy? What if mathematics is to be found in the mind/brain?^ Can this be construed as discovering it? I am not quite sure about this.

I am not sure because this is where mathematical truths and objects are thought to be by some of those who claim that mathematics is invented. This is mathematics with its truths and its objects are produced by the mind. So, it might be better to say that mathematics is created rather than invented, but I think invented is the more popular of the two terms. One question asked is if mathematics resides in the mind can it be objective (one bonus of the realist position)? 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. After all, an item in our everyday world could be considered as part of the public domain. A somewhat halfway position is that it becomes intersubjective when it is shared between minds. Some claim the objectivity of our perceptions of the universe are really at best intersubjective anyway.

So, mathematicians have a choice—either mathematical truths are real in the sense that they exist whether or not any mathematician has discovered them or not, or mathematical truths are invented (or created). Is there no other choice? I think there may just be; it is possible to construe mathematics as being constructed.† I do not mean in the constructivists sense. The constructivists only accept mathematical proofs that actually construct the mathematical objects of interests, hence there is no relying on things like infinite numbers, limits, or non-finitary proofs. I mean constructed in the sense that the mathematician constructs a proof—any proof. This I think conforms to it being discovered or invented because either way the proofs of mathematics get constructed.

Does construction then solved this issue between discoverers and inventors? Probably not to the satisfaction of many mathematicians. But, more importantly does this show that mathematical realism is a not a valid ontological¹ position? At the very least it calls it in to question.

So, what is my verdict on mathematical realism? It goes the way of Platonic realism. I see know way of knowing that mathematical realism is true, so seeing how the onus is on those who posit entities to prove that they exist (it is the same problem for those who claim that god exists) and having good reasons for a naturalist’s position, the likelihood of this type of realism to be correct is slim enough to be ignored. Anyway, I think mathematicians can get on with their work without a separate mathematical realm (they would have to if it did not exist). And, some claim that mathematics is what mathematicians do.

Next up: scientific realism. So why do people claim this type of realism, and how does it differ from the first two? For the why, I think that this type of realism survives Ockham’s Razor, so this position becomes the best explanation for the existence of things.² In other words it is simpler to claim only one world where things exist. Just like it is simpler to leave god out of any explanation. What does this simplification prove? Nothing, but it does indicate where the best explanation is likely to stand—it is the most likely to be true. How it differs is that this type of realism is found in the physical universe itself.

So, what kind of entities, without any unnecessary ones, does science posit to exist? But, first what about our everyday perceptions? It is easier to believe that our brains create these perceptions base on the information coming into our sensory organs than any other explanation. Science has done a decent job of explaining what is going on in our nervous systems when we perceive things. And, before brain science was on the scene, autopsies show the existence of our internal organs, of which the brain is one.

Then, with microscopes it was possible to see things that we could not with the naked eye. Both those things we see with our own eyes and those things we see with the help of instruments of magnification allow us to see things we believe to exist without much theoretical knowledge. There are telescopes too, but with them we look at big things with significantly more theoretical knowledge. So, what about those things that are deduced as existing. Atoms once fit this bill. It helped to explain chemical reactions, though at the time no one had seen an atom or its actions upon other entities. And, it seems that atoms were necessary to explain the phenomena of heat and entropy (the tendency in a close system for disorder to grow). Well, all fine a good, but what about direct observations of these posited atoms? Einstein put the existence of atoms of on firmer ground. He explained Brownian motion (the movement of ink in a solution) by the action of atoms jiggling the ink around. Close right?

Yes, but science could do better with the help of quantum physics, which Einstein help to produce in the beginning by showing that light could come in packets—a quantum of light—building on the work of Max Planck, who showed that energy only showed itself at discrete levels. Earlier, from other experiments, it was deduce that atoms were not the basic entities that they were thought to be. It turns out they had a nucleus, which were surround by elections (originally thought to be in discrete orbits). Then, experiments start to crash atoms and their constituent particles together in accelerators. With the help of cloud chambers, and later bubble chambers, experimenters could see tracks laid down by these particles. Each particle created its own unique track. With higher and higher energy levels the protons and neutrons that made up the nucleus of an atom were shown to contain particles of their own. These were the quarks.

So, science posits all these particles, and there were plenty of them (about 200) until Murray Gell-Mann and others tame this “zoo of particles” as it was called. There were now just six quarks which make up what are called baryons, and three different singular particles, like the electrons, which are called leptons, making up what are known as fundamental particles. Anyway, even though this seems like a lot of entities being multiplied, it was the simplest system (now called, along with the four fundamental forces and the particles that mediate them, the standard model) to be devised that would cover everything that was currently known. To posit other entities that do not make up the physical universe with its forces is to multiply more than is necessary.‡

Okay, but has anybody actually seen an atom, if that be necessary to prove its existence (which it does not)? Well, they are now able to trap such things as atoms and electrons and have visualized them in color generated images. This is as close as we have come, at least visually.

As you might have guess I see scientific realism as a valid way to look at our universe. There is probably plenty to quibble with here, but I challenge anybody to have a simpler explanation of the way things are and that they exist. I discount idealism, where everything is in the mind, as it is posited without proof. With idealism you have to decide somehow, what mind contains what, and what mind might contain it all. The usual answer is that it is the mind of god. Well, if you posit this mind, you have just done an unnecessary multiplication of entities.

Okay, now for the last version of realism I want to discuss—artistic realism. So, what is this type of realism? It is the attempt to create a piece of art (e.g. statue, painting, etching) that looks as close to what we perceive it as in life. It is opposed to much romanticized artwork. Some examples of this type of artwork is British landscape painting and impressionism. And, then you have modern art and totally abstract art.

Here is a mosaic of various examples of realism in art:

[I will remove any of these images if they are found to infringe on someone’s copyright]

Going clockwise they are: *The* S*chool of Athens* by Raphael, *Lady with an Ermine* by Leonardo Da Vinci, an example of greek statuary, *The Thinker* by Rodin, *Statue of David* by Michelangelo, and *Daniel in the Lion’s Den* by Peter Paul Rubens. My favorite part of *The* S*chool of Athens* is Diogenes* *reading a book in a beam of sunlight in the middle of the picture. And, *Daniel in the Lion’s Den *has always been one of my favorites.

While other forms of art have some interest to me, I am very partial to realism in art. I have been an abstract artist** myself, so obviously I have an appreciation of other forms of art. My least favorite form is impressionism. All art I think is a form of self-expression. Something of the mind of the artist is expressed in a piece of artwork. Other than this there are lots of different views on what art is. One definition is that art is the expression of beauty. But, beauty is even harder to define than art.

I would like to mention surrealism before I leave this discussion on artistic realism. It is very similar to regular realism in art. The pictures have realistic looking objects, but are skewed in some manner than you would see them as in reality. My two favorite surrealistic artists are Salvador Dali and M. C. Escher. A lot of Escher’s work does not involve realism, some only relate to different aspects of symmetry with non-realistic looking objects. Surrealism incorporates the blatant violation of reality, but other than that the attention to detail is the same as realism.

Here is another mosaic of images from the surrealists camp (the top two are by Dali, the side and bottom are by Escher):

[I will remove any of these images if they are found to infringe on someone’s copyright]

I have mentioned beauty above, and I would like to end on mathematical beauty. While it is not directly related to mathematical realism, I feel that most mathematicians have a sense of mathematical beauty. This maybe more so for the mathematical realist, or so I imagine. Does it give anymore support to it? Not really, but I am sure that there are arguments for it that include beauty as a component.

To end I will give my verdict on the various types of realism I have discussed in this post:

- Platonic realism – false.
- Mathematical realism – false.
- Scientific realism – true.
- Artistic realism – neither.

¹ Dictionary.com defines ontological as: “of or relating to ontology, the branch of metaphysics that studies the nature of existence or being as such.”

² http://math.ucr.edu/home/baez/physics/General/occam.html has a good summation of this princple. However, its common formulation as “entities should not be multiplied beyond necessity” is said by both wikipedia and the *Stanford Encyclopedia of Philosophy* not to be Ockham’s exact formulation in any of the available texts. He did write: “Plurality must never be posited without necessity.” I feel that this is just nitpicking. Do not they mean basically the same thing? I suppose that “plurality” could be broader than just “entities.”

* Nothing known is perfect except the philosopher’s god, and this god does not exist.

^ When it comes to philosophy of mind I am a materialist. To see more on my materialist’s views look at my post – Why Are People Afraid of Their Brain?

† See Can Mathematics Be Constructed? for more of an elaboration on this view.

‡ I have described everything here as particles and forces, but the reality is a little bit different. There is now quantum field theory, which replaces quantum mechanics, and this theory posits fields, where the particles are excitations in these fields. This theory helps to better understand the universe and the way it does its thang. Sorry that I cannot do a better job in describing quantum field theory, not that I really gave a good description of the quantum mechanical picture, but I am more familiar with the latter rather than the former.

** In the past I created fractal artwork, which I suppose is an abstract art form (in another post I will explore whether or not it is art at all), even if it is based entirely in mathematics. However, computer images (any image really) can never show a complete fractal because they cannot be created to show the infinite level of self-similarity of any fractal. Self-similarity is one of the hallmarks of a fractal. All fractals look the same at whatever scale of magnification you produce them. On my home page you will find one of my images. Here is another:

The self-similarity of the Mandelbrot set is seen upon the magnification of the border, where you find buds at every level in between other buds.

Excellent post, with many insightful points.

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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Thank you for outstanding comments.

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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Thank you for your reply, Steven, containing many very insightful and interesting points.

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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