A mathematician's response to the philosophy of mathematics

Continuing my review of the frequent dire lack of quality in academic philosophy, here is a short review of the case of the philosophy of mathematics.

Looking for what others could tell about it on Quora, the closest relevant question I found there is "Why are top mathematicians not interested in philosophy of mathematics, also top physicists not interested in philosophy of physics etc? The people who are working in the philosophy of these fields are usually philosophers.

Replies there essentially repeat the following 2 points:

While these are of course parts of the answer, there is a third part of answer ignored there, which I will develop below, focusing on the case of mathematics: that the general field of foundations of mathematics (which does not cover all genuine philosphical questions around mathematics : some genuine philosophical questions are not foundational...), is roughly divided into two variants : which can partly explain why professional mathematicians do not usually engage on what is called "philosophy of mathematics".

To illustrate that such observations were already done by other mathematicians, here is a quote from Talk:Foundations_of_mathematics

"Anyone, a mathematician especially, who appreciates the “unreasonable effectiveness of mathematics” and the “unreasonable ineffectiveness of philosophy" to scientific endeavors must recognize the dangers of letting "philosophy of math" ride roughshod over "foundations of math" and as a last line of defense, of letting "philosophy and foundations of math" ride roughshod over proper pure and applied maths.

Just look at the talk page for "philosophy of math"! What a mess. Note that some of these people actually believe the destiny of science can be mastered thru verbose semantics, concepts, schema, arguments, etc. The last time I looked, the language of science was still written in mathematics. Fortunately, bullshit had not yet taken over in the math journals.

Specialists in foundations and/or philosophy of math often over-estimate the importance of their work to those in other specialties.
Now the situation is diverse, so there are also some genuine professional forms of "philosophy of mathematics"... done by mathematicians. A reference of genuine works (appearing among top google results on feb 2019) is the article Philosophy of Mathematics (2007) by Jeremy Avigad (link appearing in his "publications" list). He wrote a more recent overview of the philosophy of mathematics: Does philosophy still need mathematics and vice versa? (24 September, 2018) explaining the bright but also the degenerate side of the philosophy of mathematics in long and interesting ways. Some excerpts gathering the main ideas:
"The philosophy of mathematics reached its heyday in the middle of the 20th century, buoyed by the previous decades’ successes in mathematical logic.(...) For a subject traditionally concerned with determining the proper grounds for mathematical knowledge, modern logic offered such a neat account of mathematical proof that there was almost nothing left to do. Except, perhaps, one little thing: if mathematics amounts to deductive reasoning using the axioms and rules of set theory, then to ground the subject all we need to do is to figure out what sort of entities sets are, how we can know things about them, and why that particular kind of knowledge tells us anything useful about the world. Such questions about the nature of abstract objects have therefore been the central focus of the philosophy of mathematics from the middle of the 20th century to the present day. (...) philosophers of mathematics were chiefly concerned with the question as to whether numbers and other abstract objects really exist.
This fixation was not healthy. It has almost nothing to do with everyday mathematical practice (...) there simply aren’t that many interesting things to say about abstract mathematical objects in and of themselves. (...) set-theoretic idealisation idealises too much. Mathematical thought is messy.(...) we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty. And so the call of the sirens is pleasant and enticing: mathematics is set theory! Just tell us a really good story about abstract objects, and the secrets of the Universe will be unlocked. This siren song has held the philosophy of mathematics in thrall, leaving it to drift into the rocky shores.
In the rest of this page I will mainly focus on some cases of bullshit, and how they are not marginal. Namely, online encyclopedia articles appear to fall prey to the above mentioned unhealthy fixation ; even when the same ideas and authors are reviewed, Jeremy Avigad does a much better job at putting them in the right perspective.

The Wikipedia article has one main quality : its list of "Recurrent themes", that is interesting questions, unfortunately not including relevant links to any answers. It also gives an interesting quote from Hilary Putnam :

"When philosophy discovers something wrong with science, sometimes science has to be changed (...) but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need."
The main content appears to be its list of "contemporary schools of thought" defined by the different approaches to the nature of abstract objects. How miserable are these for contemporary times, if they are still there, which keep ruminating the relics of the history of mathematical research which led to current foundations. Something odd: one of these "schools", the "post rem structuralism" has one name (Paul Benacerraf, who indeed was a main actor of that pseudo-debate...), but only few google results : beyond Wikipedia and its multiple copies, only 4 more results could be found when writing this page (feb 2019), likely to add itself to this number. Then I'd be tempted to ask : do these "schools of thought" have any physical address, or at least statistical data on their relative popularity, otherwise would they be themselves mere abstract objects ? in which case why should I believe in their existence ? More precisely asking whether they would be indispensable to the understanding of our best scientific theories, well... I think they aren't.

So many philosophers still believe that the "foundational crisis" of mathematics remains unresolved. Hahaha. My attitude is : the usual view of something is not clear ? Well I have no time to philosophically describe its state of unclarity because I'm too busy resolving this by bringing clarity instead !
In first approximation following the working attitude of mathematicians, Platonism can be said to be the correct view, empirically supported by the famous indispensability argument (3.2 - I don't see what it may have to do with naturalism: that is just a particular case of empirical argument even if its object is some very general stuff which I further commented in another page). But for better accuracy some clarification, nuances and details would be needed. One aspect is structuralism, that is an important aspect of what mathematics is about : while individual mathematical objects are trivial and thus uninteresting, the point of mathematical theories is to describe systems which objects can form by means of structures. This combination of Platonism with structuralism is called Ante Rem Structuralism.

But for more details, let us have a look at the Stanford Encyclopedia of Philosophy. This encyclopedia seems to have good points, as already noticed with the case of dualism: it does not only contain the traditional bullshit but also some correct view that is not found in other encyclopedic articles on the same topic. Now its treatment of Ante Rem Structuralism has the kind of advantage of making more explicit... the terrible confusion that philosophers may have in mind on the topic.
First comes the "distinction between algebraic and non-algebraic mathematical theories" first unclearly described by "non-algebraic theories are theories which appear at first sight to be about a unique model" and clearer later by "The systems that instantiate the structure that is described by a non-algebraic theory are isomorphic with each other". That is what logicians call the semantic completeness of a theory. The word "algebra" has quite different uses in higher level mathematics... and I consider crucial to distinguish cases when any two models are related by a unique isomorphism, which indeed gives a natural way to identify them as "the same system", and theories for which there are multiple isomorphisms, such as Euclidean plane geometry whose models are still better seen as essentially different in spite of being isomorphic.
But, what is a structure ? Philosophers do not seem to have a clue what they are talking about.
"In textbooks on set theory we also find a notion of structure (..) But this cannot be the notion of structure that structuralism in the philosophy of mathematics has in mind"
Ah and why not ?
"For the set theoretic notion of structure presupposes the concept of set, which, according to structuralism, should itself be explained in structural terms."
Well set theory is the natural reference theory encompassing all other theories, thus by which to define systems and their structures. Yet theories and their models with their lists of structures can be also considered in isolation. One of the roles (reasons) for using set theoretical interpretations is as a way to somehow more explicitly describe these structures of systems which the rest of theories are about. Now we need to start with something anyway to explain what we are talking about, don't we ? How can you start your explanations from nothing ?
"It appears that ante rem structuralism describes the notion of a structure in a somewhat circular manner."
So indeed some crazy absolutist expectations of trying to start from nothing do not seem satisfiable anyway... still these expectations are what philosophers of mathematics put forward on why they dislike the way mathematicians understand structures, and they think that as philosophers they are much better placed than mathematicians to undertake, though not satisfactorily, to explain the exact concept of the structures that mathematics is really all about. Oh well...

The most interesting thing of the Stanford article is the whole section 3.3 Deflating Platonism which I invite you to fully read there (instead of copying it here). I agree with Tait as reported there : let mathematics speak for itself. To comment further : I see no sense asking what is the nature of mathematical objects. For me there is only one worthy answer which I gave there : that mathematical objects are objects whose only considered nature is to be exact, unambiguous. There is no such thing as a lack of clarity of the status (nature or existence) of mathematical objects in need to be assessed with respect to any a priori more meaningful standard of "reality" : supposedly external references involved by the words "nature" and "real" in this context remain out of subject and a priori nonsensical, in lack of a meaning that may be compared to mathematical objects, but there is no point to look for such meanings, because as words without definition the question of finding a sense for these, if any, is a pure convention and matter of taste, not any worthy meaningful questions. Because mathematics is about clarity and exactness, and there is no sense trying to explain or justify what is clear and exact on the basis of what is fuzzy and unclear. Seriously, who is crazy enough to go look for what may be any real (not just metaphorical) physical standards of substances or ontology ??? Nobody could ever make any clear sense of such things. At best, the naive appearance of physical reality may be considered as an interesting metaphorical approach to discuss ontological views. Trying to refer to what may seriously be a true ontology of physical stuff would bring the discussion much too far away from the topic, into a long exploration of candidate views on the philosophy of physics.

Once so (not) specified the nature of mathematical objects, comes the question "Are mathematical objects real" ? Existence is a mathematical concept, and mathematics has its own standards of existence. It has some clear answers about it given by mathematical theorems. The most important theorems relevant to the most general existence questions about mathematical objects are

Philosophy of mathematics has a big trouble : so many philosophers of mathematics either never heard of the completeness theorem, or anyway keep mysteriously ignoring it every time they discuss an issue which this theorem already resolves or makes obsolete, either issues of existence, or of the presumed opposition between syntax and semantics which this theorem makes roughly obsolete. I added myself in 2012 to the wikipedia article on the foundations of mathematics, a section indicating the crucial lessons which the completeness theorem brings to the old pseudo-debates of the philosophy of mathematics. This section is still there. But it remains unclear whether and when the rest of courses and encyclopedia articles around will take it into account by removing all the obsolete stuff.

Now in that 3.3 of the Stanford article, is the note that "Of course not everyone agrees with Tait on this point. Linsky and Zalta have developed a systematic way of answering precisely the sort of external questions that Tait approaches with disdain (Linsky & Zalta 1995)". A referenced work to which a whole further section is dedicated: "3.5 Plenitudinous Platonism". More precisely the latter presents two versions of this Plenitudinous Platonism, the other being "Balaguer’s version". Zalta is himself the creator and principal editor of the Stanford Encyclopedia of Philosophy (yes, all that !!!!) while Balaguer is himself the author of the Philosophy of mathematics article at Encyclopædia Britannica, whose very short mention of the issue indicates an absence of any difference between his and Zalta's version.
In the Stanford article which is a bit more detailed, "In Balaguer’s version, plenitudinous platonism postulates a multiplicity of mathematical universes, each corresponding to a consistent mathematical theory". Thank you but I see no need of postulation for this, since this is exactly what the Completeness theorem already says, once assumed the axiom of infinity. In more details I see another wrong aspect of his presentation of this so-called postulate : he does not anyhow elaborate on the condition for existence of a universe of a theory, which is that the theory needs to be consistent. Indeed by the incompleteness theorem, if a foundational theory is consistent then it is unable to prove its own consistency. This suggests that there has to be something unobvious in recognizing the consistency of a given foundational theory: unless formally approached by a stronger theory, it requires some kind of philosophy to get confidence in it.

If on the other hand, instead of focusing on the specific cases of "good foundational theories" with possible philosophical reasons for consistency, we want a method valid for the totality of consistent theories, to give proper meaning to the claim that any consistent theory automatically has a model, then as the only possibly suitable systematic method, we must admit having to first spend an eternity in fruitless search for contradictions of a given theory before concluding that it is consistent and thus letting our magic stick create a model of it (actually the proof of the completeness theorem involves an infinite series of such infinitely long checkings to create the model). Because, the consistency property of theories being not knowable in advance but somehow "random" in a sense well-studied by Gregory Chaitin, to dare giving systematic earlier existence to models of every theory that seems consistent just as we could not find contradictions there after a long search, would sooner or later end up also creating models for such theories which will turn out to be inconsistent, as contradictions would finally be discovered later. And that would put us in kind of a trouble.

To give a concrete example (if I don't mistake, otherwise we may need to modify this but the main idea still goes, as can be established by the method of Gödel's speed-up theorem) : if ZF is consistent then adding to it the axiom "There exists a contradiction of ZF with less than 1010100 characters" (rigorously encoded as a formula of arithmetic, making it look undecipherable to the "naked eye") the resulting theory can look pretty much consistent while it actually isn't, since a contradiction can be found by enumerating all strings of less than 1010100 characters and not finding any contradiction of ZF among them, but there cannot be any much shorter formal proof in ZF than this one of that result.

Now, regular mathematicians with their working mathematical Platonism basically don't have any problem packing up in their universe of discourse this infinite series of operations involved in the proof of the completeness theorem, but why would philosophers of mathematics like to work with that ? In his own words from the E.Britannica article, balaguer "argued that it is only by endorsing this view that Platonists can explain how humans could acquire knowledge of abstract objects". Good luck. He explains further in the second page:

"if full-blooded Platonism is true, then knowledge of abstract objects can be obtained without the aid of any information-transferring contact with such objects. In particular, knowledge of abstract objects could be obtained via the following two-step method (which corresponds to the actual methodology of mathematicians): first, stipulate which mathematical structures are to be theorized about by formulating some axioms that characterize the structures of interest; and second, deduce facts about these structures by proving theorems from the given axioms."
Yes, like I did when I undertook writing (but did not finish) a proof concerning a particular variant of the Hadwiger–Nelson problem, which is that no almost-covering mapping of the plane with 5 colors and with the Baire property satisfying the condition of that problem can exist. The study went by first postulating that we had such a mapping with 5 colors, then progressively proving more and more theorems about its properties, until a contradiction would be reached. By the way, where was the wonderful postulate of existence of all possible abstract objects by this "full-blooded Platonism" so useful in this process ? the thread of the argument seems to have been lost here.
But anyway it remains a fact that there is a known proof theory that does its job, which is that it allows by finite means to reliably deduce facts on any possibly infinite system from any given axioms, that is, any formulas which happen to be true about some mathematical systems; that, human brains being much more complex than that, while general abilities to understand the world could be naturally favored by evolution, and the world (generally more or less any world) naturally contains diverse kinds of mathematical structures among its aspects : all this naturally leads to regard the basic, general abilities to understand maths by means somewhat similar to the job of formal proofs, as totally non-mysterious independently of any choice of subtle philosophical, ontological orientation. In other words, I consider totally crazy to go imagine that, under any reasonable philosophical view (be it platonistic or materialistic...) any difficulty of accessing mathematical truth in simple, "normal" cases (letting aside subtle cases of high maths I mentioned in another page) under the excuse of the abstraction (non-materiality) of mathematical objects, would be worth considering in the first place. Any philosopher who would imagine that any mystery exists here is just dumb, relying on completely inappropriate kinds of approach based on some insane obsession with substances or the like. Therefore, as long as consideration is so restricted to basic math cases, the issue of "how to account for" this non-existing problem cannot be used as a rational argument to support any philosophical position against another either.

Checking the Linsky & Zalta 1995 article

"In this paper, we argue that our knowledge of abstract objects is consistent with naturalism. "
Since basic math abilities are devoid of mystery and thus obviously compatible with any sensible view, the whole point is void then.
More references inside the article, of how to "provide an account of our access to abstract objects" as if it was a real problem. As above explained it is an empty, non-existing problem, so that all such mentions of it as a real problem are perfectly ridiculous. Skipping these and looking at other claims.
"Naturalism is the realist ontology that recognizes only those objects required by the explanations of the natural sciences."
Nonsense. This claim is at the same time vacuous, i.e. tautological (just define the phrase "existing objects" as "whatever is required by natural sciences" = to explain facts observed in the world, and the claim becomes true whatever substantial metaphysical claims may be otherwise stated), and logically unrelated with more usual, substantial conceptions of naturalism (that the fundamental stuff of reality would be some kind of material or otherwise non-conscious stuff), so that it is totally crazy to so over-determine the meaning of "naturalism" by such a priori unrelated criteria. In other words, what would become of naturalism if experiments established the necessity of involving the influence of immaterial spirits as necessary ingredients to account for biological facts: would it proudly accept these as existing objects, or admit being falsified ????
" The problem is that traditional Platonists seem to rely on naive, often unstated, existence principles, such as that every predicate denotes a property (or picks out a class) or that a theoretical description of an abstract object is sufficient to identify it."
Indeed there were some naive mistakes by some pioneers of mathematics, who did not understand how the mathematical reality was shaped. Now mathematicians with proper education in logic know that any theory may refer only at first sight (when taken literally) to a unique model, but in fact (looking at it from the outside of the particular theory but still mathematically), it is only a possible model in a range of other possibly existing models (if the theory is consistent). This does not contradict the reality of these models as abstract mathematical systems independent of the material world.
"The logical positivists articulated this worry by arguing that our knowledge is either empirical or logical in nature and that in neither case could we have genuine, synthetic (ampliative) knowledge of nonspatiotemporal abstracta."
Hum... depends how we understand logical positivism and the kind of "nonspatiotemporal abstracta" to be considered. It is possible that some logical positivists wrote unclear or mistaken views beyond what I understand as the main and genuine rationality principles usually referred to as "logical positivism". I endorse a kind of light version of logical positivism where in the sentence "our knowledge is either empirical or logical in nature" I understand "logical" to mean "mathematical", as I cannot see significant or clear fundamental difference between logic and mathematics.
"Quine formulated a limited and nontraditional kind of Platonism by proposing that set theory and logic are continuous with scientific theories, and that the theoretical framework as a whole is subject to empirical confirmation."
Quine is free to believe what he likes and I may think differently ; I don't like his NF system which I consider inappropriate as a foundation of mathematics, and hardly any specialist in set theory is interested in it either, so why should I care ?
" Most Platonists conceive of abstract objects on the model of physical objects. That is, they understand the objectivity and mind-independenceof abstract objects by analogy with the following three features of physical objects (...) We call those Platonists who conceive of and theorize about abstract objects on this model of physical objects Piecemeal Platonists. Historically, Piecemeal Platonism has been the dominant form of traditional Platonism, for traditional Platonists typically assume that their preferred abstract objects are “out there in a sparse way” waiting to be discovered and characterized by theories developed on a piecemeal basis."
Without regarding physical objects as a "model" to conceive mathematical objects, but instead letting mathematics speak for itself, it just happens by coincidence that mathematical objects are rather well described in such ways. Then what ? This is perfectly coherent with the fact that the pioneers of mathematical Platonism and the foundations of mathematics did not immediately come up with the right, perfectly correct versions of all aspects of their respective topics, but corrections and improvements to these views had to be made later to better fit with the actual shape of mathematical realities that could be found after more detailed examination.
"one must assert topic-neutral comprehension principles that yield a plenitude of abstract objects.... Some of these principles are distinguished by the fact that they assert that there are as many abstract objects of a certain sortas there could possibly be (without logical inconsistency)...We shall argue that a Principled Platonism and philosophy of mathematics based specifically on the comprehension principle for abstract individuals is consistent with naturalism. "
I'm afraid he has no idea of the extreme discrepancy between what he claims to be doing and what he actually does. In mathematical standards, proving for example that the Axiom of Choice and the Continuum Hypothesis, or their negations, are consistent with the ZF axioms, is far from a small affair. We have to first specify the whole exact logical framework in which we work, then what we need to prove is that in this logical framework, all possible deductions from the given combination of hypothesis will never lead to contradiction (assuming that ZF alone was consistent). It requires a method to somehow anticipate and control the infinite set of all possible logical deductions from these hypothesis which can ever be made in the future (even if the actual method to ensure this does not proceed in these terms but through semantics: constructions of models...). It does not suffice to have a small look at the given statements, see them roughly agreeing on something, figure out a couple of possible deductions and not see a contradiction or disagreement simply popping up, to justify the conclusion that they are actually consistent.
First there is no such a thing as a logically clear formulation of "naturalism" (but all I see is just a big play of switching between different unclear pseudo-formulations without any decent link between those).
Second, his "Principled Platonism" while being expressed in kind of mathematical notations, direly lacks clarifications about the meaning of each of their ingredients and the logical framework in which these ingredients would be so put together to form such kinds of formulas.
Third, no attention is paid to the consistency of this all with the actual contents of mathematical realities that they are supposed to account for.

Namely, while I am quite familiar with mathematical logic and of course not afraid of mathematical notations, the formulas in his principles look like a big bunch of nonsense visibly impossible to decipher as they are senseless anyway : just like it is possible and even sometimes appropriate to express some very rigorous mathematics using ordinary language, it is also very possible to express awful nonsense in mathematical symbols, a trickery I already noticed in some other case of pseudo-science. I guess he actually does not give a shit about any consideration for mathematical rigor or consistency in the formal expression of his principles, under the excuse that he is not working inside mathematics but supposedly expressing some kind of extra-mathematical claim, through which he thinks all mathematics could be founded on some extra-mathematical basis. That reminds me the trick of Christian apologetic, that consists in providing bunches of flawed, pseudo-rational arguments and at the same time claiming to be all about stuff beyond the mere "flawed" human abilities of reason, as an excuse to avoid the burden of having to defend the validity of their arguments in the face of rational criticism.
All I can do is skip unclear or possibly nonsensical stuff to search elsewhere for possibly substantial claims. And as far as it seems substantial, his 3rd principle seems totally inconsistent with some basic maths :

"3. If x and y are abstract individuals, then they are identical iff they encode the same properties.
Let us take the example of 2 points in an Euclidean plane. Points are abstract individuals. In Euclidean geometry, every point has all the same geometric properties as every other point of the same plane (or of different planes, which we may need to distinguish for some reason). But if for this reason you decide to see all points as identical (a single point) then you collapse the plane to a single point and there is no Euclidean geometry anymore. I know, I know, it is the fault of all these awful school teachers who trained their pupils to lose any idea of the concept of point in a plane and replace it completely by that of oriented pair of coordinates... but why did some philosophers of mathematics never care to grow up from this shit since then ???
" The comprehension principle asserts the existence of a wide variety of abstract objects, some of which are complete with respect to the properties they encode, while others are incomplete in this respect. For example,one instance of comprehension asserts there exists an abstract object that encodes just the properties Clinton exemplifies. This object is complete because Clinton either exemplifies F or exemplifies the negation of F, for every property F. Another instance of comprehension asserts that there is an abstract object that encodes just the two properties: being blue and being round."
Hum, in pure mathematics I know of no such property as "being blue". I guess one may safely say that Clinton is neither blue nor round, but can it really be said that Clinton really either exemplifies F or exemplifies the negation of F, for all other properties F as well, such as for example the property of being "complete" (just to pick an example of property named in that excerpt) ? and I once saw a report that different cultures (human languages) give different limits for giving names to colors, therefore raising some translation issues. As for "being round", well there is the property of "being a circle" in Euclidean geometry, but I doubt it may qualify any non-mathematical objects. As for "being round", well, where exactly is the limit supposed to be ? Being round is part of the definition of a dwarf planet. Ceres is considered round enough to be a dwarf planet, but Vesta isn't. I am not trying to start any controversy on these particular cases, but just to wonder how should we cope with these definitions if some intermediate case happened to be found.

Somehow a main idea emerges from all this pseudo-formalism : he starts from "properties" coming as never defined words of common language, but just supposedly qualifying (the heck knows how) some things that appear from the world, but themselves never defined either (as no care is given to analyze any ontology of physical or otherwise supposedly "real" stuff through some study of theoretical physics and choice of interpretation of quantum physics...). Then he defines "abstract objects" as any kind of pack of such properties whatsoever, without any distinction. In particular, under this conception, any pack of mutually inconsistent properties will form some "existing" abstract object just as well. This finally reveals what sense he is giving, in his mind, to the concept of mathematical object : he is conceiving these objects as nonsensical packs of nonsensical properties of supposedly real but never defined or analyzed external stuff. A nonsense over a nonsense over a nonsense. All the opposite, thus, of my above definition of mathematical stuff as clear, exact stuff. I understand: like all philosophers who are not mathematicians, there is no way he can ever admit the possibility for mathematical objects, which he is unfamiliar with (since he is not a mathematician) to be actually making any clearer, more meaningful or more consistent sense than all the more directly visible stuff from nature, which seems more real to him since that is what he is more familiar with.

"Consider a mathematician who at one time accepts the Continuum Hypothesis (CH), and then later rejects it, or consider two mathematicians who disagree about whether it is “true”. We claim that they are thinking about different objects—they just don’t realize it. "
Please give names, or are you only talking about abstract hypothetical mathematicians from your imagination ? as I can hardly think of any concrete examples.
More seriously, consider that over 99.9% of mathematicians only work with set theory in ways which happen to be conclusive while never having to mention nor even less use either CH nor its negation - for the good reason that all usual mathematics is expressible inside Second-order arithmetic, while CH isn't. Are they failing to realize that they are talking about different objects just because the one's work is valid under CH (which he neither mentioned nor used) while the other's work is valid under the negation of CH (which he neither mentioned nor used either) ? But if they are talking about the same objects just because their discourse is independent of CH, then are these the same or not as the objects of the study of a mathematician who does use CH, or to one who does use the negation CH ? But if the objects of the first are the same as those of each of both latter cases, then how can these not be the same as each other ? What about the one who comes to review both cases to reach a conclusion anyway valid in each case for a different reason ?
Consider that CH is not the only undecidable statement for which this happens, but it is just one of infinitely many similarly undecidable and mutually independent set theoretical statements, a list which by necessity has to be open-ended since there has to exist such undecidable statements whose undecidability is itself unprovable (there cannot be any systematic way to discern which ones are in this case). Would mathematicians, obviously having to work with incomplete set theories and rarely dealing with any undecidable statement, never be talking about the same thing to each other just because their discourse remains valid in an infinite diversity of universes ? But they are aware that there is such an infinite diversity of universes in which their work remains valid !!!

In some kind of concluding part of the article he writes

" But the most important ingredient of Platonized Naturalism is the argument that Principled Platonism is consistent with naturalism. Indeed, from a naturalist perspective, how can there be synthetic a priori truths like the comprehension principle? The answer is that there can be such truths if they are required to make sense of naturalistic theories, that is, if they are required for our very understanding of those naturalistic theories. To establish that our comprehension principle is required in just this way, we offer a very general argument that begins with two premises "
again with this nonsensical definition of "naturalism" as some kind of epistemological tautology (basic empirical move). The first premise is some kind of triviality (or maybe not but it does not really matter).
"the second premise is simply that the comprehension principle and logic of encoding are an essential part of the logical framework required for the proper analysis of natural language and inference. In support of this premise, we claim that the comprehension principle and logic of en-coding are the central components of an intensional logic that offers the best explanation of ..."
yes yes he is just here to boldly claim that his own ideas are the best explanation of everything, and that is his whole argument. Of course like everyone he is free to think that his own ideas are the best, and I understand that he may really believe so, as if he did not then he would have most probably adopted some other ideas that looked better to him instead.
But to me it rather all looks like one of the worst explanations of the nature of maths I ever saw (uh maybe because I am not generally familiar with such bad ones but...). Because all his arbitrary incoherent postulates, instead of providing any effective account of mathematical existence, are following the lazy option of essentially redefining the extra-mathematical "existence" predicate over abstract entities to actually mean the constant true predicate, which just gives equal "existence" to any thinkable or unthinkable, consistent or inconsistent, possible or impossible stuff (except for the coexistence of distinct abstract objects having all the same properties). But such a vain play of vocabulary, emptying the word "existence" from any original meaning it might have (I'm not here claiming it had any meaning before, but only noticing how well he ensures, by this redefinition, to empty this word of any possibly effective meaning), cannot suffice to resolve any substantial problem or can it ? Namely:
I don't see how giving such an equal "existence" to incoherent as to coherent "abstract objects" may explain how properly rigorous mathematics could turn out to be more useful or relevant to the description of nature, than random, incoherent bunches of pseudo-mathematical formulas.
But even if that could be accounted for, I have even bigger difficulties with how it might explain the famous unreasonable effectiveness of mathematics in physics, namely (I must explain for all those philosophers who have no clue how this report can be serious and actually mean something meaningful), that some very deep, mind-blowing, highly abstract mathematics which somehow fits very much the minds of those who can grasp it, which come to the inspiration of mathematicians based on purely theoretical elegance criteria very far from anything which resembles daily life, could turn out be the key to the formulation of theoretical physics, unlike some more earth-bound (boring, endlessly iterated and patched basic-style school level) style of mathematics which, actually, more properly qualifies the laws of biochemistry and everything that can be built on top of it, that is ultimately all what matters to the criteria of natural selection which supposedly drove the development of the skills of human brains (while the mathematics of theoretical physics leaves almost no concrete trace at the level of natural selection processes).
Even more specifically, his 3rd principle stating that any two abstract individuals "are identical iff they encode the same properties" seems to me the worst possible explanation for why the kinds of mathematical laws which were discovered to wonderfully account for particle physics (the standard model) are mathematical laws displaying very rich continuous groups of symmetry (gauge invariance), which precisely means that they have lots of kinds of continuously many different abstract objects which have all the exact same properties.

Now more globally, what is the main structure of that "work" ? To first make a fuss about one crazy view named "Naturalized Platonism" developed by some other dumb philosophers ; to point out that this view had some defects (indeed). Then to present his own, different crazy view (Platonized Naturalism) and to feel proud reporting that his new view does not display the same defects which the previous crazy view had. So, just all about 2 crazy views accidentally picked up out of thousands of possible crazy views (that he does not have himself the imagination to think about, since he just happens to be so focused on these two), which other stupid philosophers might imagine as well, while each of which can surely be shown to have a similar advantage of not having some precise defects which could be pointed out in some other crazy view.
And what is the motivation of it all ?
We must understand the difficult situation and growing concern which many philosophers of mathematics are facing. While their work is supposed to be all about accounting for the nature, foundations and success of mathematics, it turns out to run into a worse and worse discrepancy with the actual works, practice and views of mathematicians themselves ; growing discrepancy that they are more and more desperate trying to resolve or even account for. This make them eager to dream about and attempt to propose any kind of resolution that looks good. Yet the solutions they can search for, propose and approve, cannot and will never be any real solution, that is solutions that can actually look good in the eyes of mathematicians. Because solutions that can be real or actually acceptable by mathematicians, would be both out of the reach, and none of the concerns, of philosophers. Because their only concerns are to entertain their dreams (illusions) of resolution, and pass peer-review where peers are not mathematicians but peer philosophers, themselves similarly focused in their illusory goals and methods, with their minds deeply rooted in the same general misunderstandings on general ontology, the nature of mathematics and so many mistaken ways of thinking and questioning, which philosophers kept entertaining in their long traditions.
And to desperate problems, desperate solutions. In the face of the triumphant working Platonism of mathematicians which they cannot account for nor even decipher (while they do not notice their failure to decipher it as they stay faithful to their traditional misunderstanding of it), the kind of "solution" thus proposed by the tenants of Plenitudinous Platonism may be described as follows : let us take the clothes of the super-nice guys ! We cannot understand the triumphant mainstream position ? Let us just offer a statement of extremism in its support (or at least in the support of what we think it is) ! We could not succeed to criticize the king ? Let us become more royalist than him ! So undertaking to behave as the mathematicians most faithful dogs eager to bark at any stranger without any sense of appropriateness, can anyway help us look brilliant in the eyes of our miserable peer wolves who were becoming desperate at how to resolve their conflict with humans.

About rival foundations

Still in the Stanford article, is the remark:
"The thesis that set theory is most suitable for serving as the foundations of mathematics is by no means uncontroversial. Over the past decades, category theory has presented itself as a rival for this role. (...) Unlike in set theory, in category theory mathematical objects are only defined up to isomorphism"
No they aren't. Category theory is just an approach in which the concept of isomorphism is given crucial importance. And while some category theorists may have considered category theory as a "rival" to set theory, such a view of these as "rivals" doesn't make any sense. Set theory and model theory are both steps in the cycle of mathematical foundations, each one being the natural foundation of the other. The concept of category is also very important somewhere in this cycle. It plays a crucial role, but is no rival. How could it be ? Some people don't like set theory because set theory is usually considered as not very friendly to the concept of isomorphism (while the above commented Linsky & Zalta article is so proud to sum up its whole understanding of mathematics to the claim of non-existence of isomorphisms just because that is what ZF teaches). But why ? Just because its traditional formalization, ZF, does not accept urelements nor functions as primitive objects, but only sets, and so builds all needed objects as sets, thus sets of sets and so on. However it just happens to be the traditional formalization of set theory, but not the only possible one. Why is it the traditional formalization, which everybody usually keeps following ? Just because it is supposed to be more elegant to have only one kind of objects, which are sets, and let all other objects be built out of this. It is usually supposed to be more elegant, except when it is not. So when it is no more, it can be time to consider a set theory with not only sets. There is no need to throw away the use of some set theory altogether, for the profit of some fanciful presentation of category theory as a starting point of mathematics, which as far as I could see, cannot look like any decent starting point of mathematics at all.

Review of "A Formal Apology for Metaphysics"

Just an article I stumbled upon, also dealing with connections between maths and metaphysics... reference (discussion).
Again, philosophers are on the defensive in the face of the evidence of the superiority of mathematics over their activity. Some arguments there may be defensible. However it has a huge flaw: on the most important gap justifying to see a huge difference of value (in my view), he tries to deny the gap, by telling big nonsense.
The huge difference I see is in terms of "security" (his 3.3). He grossly underestimates the security of mathematical knowledge. This has itself 2 parts. In the first aspect, this is BS:
"A theorem may be provable in, say, classical logic, but fail in intuitionistic logic or paraconsistent logic. Because there is a substantial debate over what the ‘correct’ logic is, it follows that such theorems may not be entirely future-proof. Developments in philosophical logic, or indeed in meta-mathematics, may alter the landscape of what follows from what in a substantial way".
No there is no such debate, and the issue of what follows from what cannot be altered. Classical logic (first-order logic) is the clear standard reference of what math basically means, in which all theories are normally understood (there is the other reference of second-order logic, where some things become more fuzzy, but this no way undermines the validity of theorems that were obtained from first-order logic). Theories in the ordinary sense must of course assume it to make the sense they are there for. You may of course choose to play nonsense by adopting another "logic" which denies the validity of proofs of classical logic, and consider "theories" in these other "logics" but even if you have the crazy idea of introducing there a "theory" by writing there "the same axioms" as a usual theory normally using classical logic, it won't make any more the sense of the initial theory, so that it is out of subject for what the initial theory was about.
Second, is the question of the justification of axioms of set theory (which is the one theory which must be accepted as general framework for mathematics). Basic axioms (ZF), which largely suffice for most purposes, are actually reliable (while the big debate on possible additional axioms hardly interests any mathematicians outside this specialization). The only significant controversy is about the axiom of choice, which is usually accepted, though it only matters for properties of quite abstract infinite systems beyond normally applicable maths. I admit this underestimation of reliability may be an honest mistake by lack of clearly written public knowledge of such justification, a gap which my work on the foundations of maths comes to fill.

Then the gap is huge between the security levels of maths and metaphysics for the following further ignored or under-estimated reasons

Finally, a debate can be done on the applications of metaphysics, but I have no time to address this now.

In conclusion

The resulting state of affairs, from a mathematician's viewpoint, may be described as comparable to what a modern biologist would feel if, looking at reference works on the philosophy of biology, he would find there a list of competing positions presented side by side on an equal footing, among which : a badly summed up account of evolution theory inspired from Darwin's book on the Origin of species ; Young-Earth creationism ; and the theory of the flying spaghetti monster presented as a serious candidate theory without any sense of irony.

Alain Badiou

He appears famous in philosophy and refers much to math and set theory; however as I could see no indication that his writings make any sense (except when they are just a faithful copy of works of mathematicians, of course, while his particular comments there, such as his taste for surreal numbers over other numbers systems, are mere unjustified fantasy), I do not consider him worth much of my attention, so I'll rather give here some links to comments written by others. Others found it nonsense. Another comment. He was object of a hoax similar to the Sokal affair : article in French - other article. As for Badiou's discernment abilities in matters of philosophy and politics, well, you can figure these out on that little beginning of article.

More external references

Other Quora question: What do mathematicians think of philosophers of mathematics (and the philosophy of mathematics that come out of it)?
A long report of a poll of mathematicians on Platonism and other philosophies of mathematics (it is very long so I did not read it)
Barry Mazur (mathematician) wrote diverse philosophical articles, such as Mathematical Platonism and its Opposites
A defense of mathematical Platonism

Related pages in this site

What is science
Philosophy : currently a pseudo-science
On the philosophical treatment of dualism
About the FQXI essay contest on the math/physics connection
Missing the real core of Science