Quora Answer: Do Continental philosophers resent their lack of mathematical ability?
This is a silly question. This is the kind of Question that we should not answer. If we were to not answer questions like this then perhaps questions would improve here on Quora, where there are many silly questions and the only reason that quora is worth reading is the fact that good people who know something take time to answer these silly questions.
But since I don’t want my complaint about this silly question, whose author cannot be serious about, I will offer an answer myself, even though it is against my better judgement.
First of all Continental Philosophy is just the Western tradition carrying on. It is Analytic Philosophy in England and the USA following suit which branches off into Realistic Empiricism, Language Critique, and Language Games, mostly specialized argument for arguments sake.
Since Math is the basis of all philosophy, Continental Philosophers are at home in it as any western intellectual and their work is meant to deepen our knowledge of science, just as analytical philosophers do. Neither specialize in Mathematics, because Philosophy is more fundamental than mathematics in many ways. Plato said that mathematics was a pre-requisite for learning about philosophy, and Continental Philosophers take that just as seriously today. First of all we must realize that education in Europe is superior to education here in the USA. So they know more math on average than equivalent philosophy students in the USA. So the whole premise of the question is just silly. And is a joke, and thus we should not take such questions seriously. When we say this we are denigrating Europeans in general when in fact their standards are higher than ours in many cases, and their students certainly know more than ours at given levels of education. Their Ph.D. degrees are more valuable than those in the USA because they have higher standards of scholarship in many cases.
So now to the serious answers that have been made to this question. Mention is made of Badiou’s interest in Set Theory as the basis of Being, and Deleuze’s use of Calculus as a reference in his philosophy. Of these the case made by Badiou is the most interesting. However, it is Deleuze’s philosophy which is deeper of the two, which would be the case whether or not he used Calculus as a grounding for some of his ideas. Badiou’s idea is interesting because of its extremity. But also because his argument centers on Cohen’s Forcing argument within set theory. Badiou makes a clever generalization of Cohen’s argument. I made a lot of progress in my dissertation research by showing why Badiou’s argument is incomplete. Let me mention just one way.
Badiou says Set Theory is Being. The one thing I can identify with in this idea is that Set Theory is based on axioms and the axioms are fragmented as shown by Cohen’s argument and so that generally shows that Being is fragmented which is a position I have held for a long time, which is contra to the Idea that Badiou criticizes of Deleuze that Being is Univocal despite its embrace of heterogeneity. Identity and difference are the two sides of one aspect of Being. There are three others: Reality, Truth and Presence with their anti-aspects. Deleuze squarely stays within the perview of ontology by claiming that Being is Difference but has a de-totalized totality (to borrow a phrase from Sartre) that is univocal and thus conferring unity of a sort to Being. What Badiou is missing is that there are many claimants to the throne of being the basis for Mathematics and Set Theory is only one of them. We have to consider the others, like Category Theory for instance, or Mereology, etc. But beyond this there is another deeper problem. Sets have a dual, which are Masses. Masses appear in our language as non-count nouns as opposed to count nouns. Non-count nouns are things like furniture or grass, we say as piece of furniture or a blade of grass, i.e. we have to have a counter to refer to a part of the mass. Masses cannot be counted unless there is a counting term. Now from a Category Theory point of view the anti-set you get when you reverse the arrows are also just sets. So something more than merely reversing the arrows of the category Set is needed, and I call this an inverse dual. Dual would just be reversing the arrows, but also some properties have to be inverted as well. For instance the emergent part of the set is the particular within the set not the set itself. On the other hand for Masses the emergent part is the whole mass and not the instances that make up the mass and exist within its vocabulary. But more importantly Sets have Syllogistic Logic while Masses have Pervasion Logics like that developed by G. Spencer-Brown in Laws of Form which was clarified by N. Hellerstein in Diamond and Delta Logics. So when Badiou goes on to describe Worlds in terms of Logic, we can be sure that he is talking about syllogistic logic which is the only one we know here in the West. But pervasion logics were the more prevalent in India and China. And so although I have not read the second volume of his study yet, I will bet that he does not deal with Pervasion Logics just like he did not deal with Masses. So this is a major flaw in his attribution to Set theory of all the trappings of Being itself, because there are obviously masses with Being as well as particulars that can be placed in Sets. So it is clear that Badiou is wrong, in general, but the fact that he uses higher set theory as the basis for his analysis really does raise the bar on theories of Being, because there is a sense in which what he says is true. That is Set Theory has some key characteristics that are of necessity the same as those of Being like perdurence. But of course this perdurence is bought at a price and that price is the fact that it occurs in a void, where there are no particulars yet in the sets but the sets are pure projections with the null set and empty set acting as marked and unmarked signs. The appearance of the first particular he calls the Ultra One. Prior to this are two principles: The Event, and the Multiple. These prior grounds of the Set as empty projection are prior to One arising, so the Multiple comes before One and Many. It is what Badiou calls true heterogeneity and difference (unlike that of Deleuze) without reference to the One either as totality or unity.
What I learned from Badiou’s use of Cohen’s forcing briefly is that a transcendental is the same across an entire immanence and that is why it is an invisible, because it is something that does not set up a difference within the immanence. Forcing does outside the immanence and forces a transcendental on it to see if there is a difference generated. It there is no difference generated then it is the same transcendental, if there is a different difference generated then it is not the same transcendental. This may be a wrong interpretation but this is what I got out of what he was saying generalizing on Cohen’s work and seeing ontological significance in it. It has long been assumed that transcendentals are infinite. But in Cantor’s theory we can only really tell the difference between the countable and uncountable (real number) infinitudes. What is strange is that cardinal infinities are not at set distances from each other beyond aleph. This I think shocked Cantor and others once they got use to there being different infinities. We are used to numbers being about the same distance from each other in normal math. When suddenly we don’t know the distance between them then we really don’t know what it means when we count to higher infinities. It more or less makes the exercise fruitles.
If a transcendence is an infinity, and then we try to go to the next higher infinity level to get a higher transcendence, if that movement does not make a difference within our immanence then really we have the same transcendence as we started with. As we expand the Universe of coverage by transcendences then we are forcing certain properties on the immanent realm that is being immersed in the larger space. If this forcing does not produce a difference within the immanent range then there is no difference that makes a difference by the introduction of the new transcendence, and really all we have is the old transcendence in a new guise.
This is my understanding. I stand ready to be corrected. If I got anywhere close to the right answer I am happy because this is one of the most esoteric ideas I have run into. But it allows us to know that the Continuum Hypothesis is independent of the ZF axioms, which means the fragmentation is real between the ZFC axioms.
For me this was important for my research into Schemas Theory because after formulating General Schemas Theory I thought I had to then formulate a worlds theory in order to know its context. General Schemas Theory gives the context of Systems Theory, or the theories of the other schemas. But what I realized is that I did not have to give the context of General Schemas Theory if the world theory did not produce any difference at the level of schemas. For instance we produce General Schemas Theory so we do not disturb the meanings of lower level Schemas like facet, monad, pattern, form, system, meta-system or domain, etc. If the world theory we introduced affected the structure of Schemas Theory, or if Schemas Theory produced a difference in systems theory then we would think it was an anomaly and would try another approach toward producing a theory that was truly general.
Once I realized that any World theory I created would not change general schemas theory, I stopped trying to find a context for Schemas theory. It saves us from needless foundational searches such as I was engaged in. It does not matter which world we are in if it does not change our schemas, and it does not matter which schmatization we have if it does not change the various schemas that are covered by the generic schematization. This turns out to be a good test of generality of a theory. We don’t want ad hoc changes by introducing a new level of transcendental.
This turns out to be an important metaphysical consideration. And we should give Badiou credit for understanding that there was a metaphysical equivalent to forcing.
Mathematics is odd. Analysis banned infintesimals, and this gave rise to non-standard analysis. Yet Mathematics accepted Cantor’s paradise. And we justify our Idealist aspirations by the fact that we can understand infinities as processes, even if we cannot understand them in terms of Pure Being, as a frozen fully present reality. There is something strange going on here, which also implicates the split between intuitionist or constructionist mathematics and traditional mathematics that subscribes to excluded middle. The mathematical universe is not symmetrical. For instance we only have topology and geometry as mass maths while almost all of the rest of our math is set based, and that is why we might be fooled into thinking that Set Theory is all we need to ground mathematics. Husserl developed his entire phenomenology in order to answer the question “What is a number?”. When we see the complexity of consciousness seen from a Phenomenological perspective it is amazing that we can do math at all.
I have shown in other writings that there are various forms of math associated with the Meta-levels of Being.
Pure Being = Calculus
Process Being = Probability
Hyper Being = Fuzzy Possibilities
Wild Being = Chaotic Propensities
Ultra Being = Singularities in Catastrophe Theory (Rene Thom).
But when we look at mathematics what we see is that it is really all Present-At-Hand or inscribed in Pure Being. Even Probability theory is described in terms of functions. So I have hypothesized not only that there are mathematical forms at each meta-level of Being, but also that these should be split, perhaps asymmetrically between Set-based and Mass-based Categories. However the current asymmetry strikes me as being too great.
Fundamentally I think that Constructivism and Intuitionism is more correct than Traditional Excluded Middle mathematical argumentation because it is precisely the excluded middle that bans non-duality.
I am putting my money on Surreal Numbers as the more basic kind of mathematics based on Game Theory. From it you get all the kinds of numbers that otherwise have to be constructed based on the limitations of each earlier kind of number. Surreal numbers gives us all the various kinds of numbers, along with both Infinities and Infintessimals. But unfortunately it is meta-systemic and thus the numbers have holes in them, and you can calculate with the holes. It as the void prior to the arising of the progressive bisection of the number, and then there is the emptiness between the braches of the number tree where a specific number is approximated by the up and down arrows that represent the moves of the game within the field of all possible moves.
Plato says that all who enter his academy should know geometry. Mathematics is the bar that is placed at the entry that must be passed to get to the real work of understanding the Platonic Forms. And there is no greater example of the representable Forms than all the various categories of mathematics discovered in the last century which has been a renaissance for mathematics. Who would have thought that we could finally prove that we had come to the end of the sporadic groups. Who would have thought we would have come up with not just category theory but n-category theory, or topoi, or discovered that the fourth dimension had no stable topology (Donaldson). The list goes on and on as the new mathematical categories have overwhelmed us as if we were caught in an avalanche of representable intelligibles. But what progress have we made on the representable intelligibles? Not really very much. Analytical Philosophy spent much of its time trying to prove that philosophy itself was just a bunch of mistakes following the language philosophers and Wittgenstein. My favorite of the lot was Schlick who sponsored Wittgenstein as the spokesman for the Vienna movement (Notice this was on the Continent.) Frege (who also lived on the Continent, and) who criticized Husserl’s early phenomenology and who changed his position based on Frege’s critique of his dissertation. Yet Husserl is forgotten by the Analytical Philosophers. Funny how Analytical Philosophers have such a bad memory. They cannot seem to remember anyone after Frege who lived on the Continent except Wittgenstein. Then suddenly everyone from then on lived in England or the USA. Meanwhile back in Europe, Europeans carried on Philosophy as they always had remembering Hegel, Kierkegaard, Nietzsche, and all those in the Phenomenological Tradition after Husserl, like Heidegger, Merleau-Ponty, Sartre, Deleuze, Derrida, Lacan etc. Continental Philosophers are just plain old Philosophers, while Analytic Philosophers are actually the ones with the resentament. First of all their arguments are isolated among themselves and have no bearing on Philosophy (Continental Philosophers are superstars in their own land compared to American or British philosophers, mainly because what continental Philosophers have to say are normally relevant to what is going on within society, culture, contemporary events, politics, etc.) And as I have shown in other posts Analytical Philosophy is dying while Continental Philosophy is thriving even if philosophers in general are not doing so well of late. So if you are in a field where you have to send out letters to students that say there is no chance you will ever get a job if you do a Ph.D. in Philosophy, which is irrelevant to contemporary society or culture, and which is being mentioned less and less in books, then that is the real case for resentment. Especially if you know that your continental friends are better educated and their Ph.D. programs are more rigorous than those in the USA. Do we sense in the animosity of the Analytical Philosophers some chagrin at being part of a dying field while Continental Philosophy has been adopted as the saviors of English Departments, giving them something worth while to talk about finally. And there are just so many more English students and departments than there are those in philosophy. And the whole sale adoption of Continental Philosophy by English Departments as their basis for the criticism of literature of all kinds, means that there is an unstoppable expansion under way for Continental Philosophy. They do not tell their students that they will not be able to get jobs as English majors. They tell them that they can write books and become famous authors and make lots of money or get a cushy job in any number of schools, colleges and universities that all teach English, even as a Second Language. In other words Continental Philosophy is taking over the world and becoming a dominant paradigm for looking at our own society, culture, literature, media, politics, psychoanalysis, psychotherapy, etc. Analytic Philosophy the child of McCarthyism is fading away slowly but surely, because they have nothing interesting to say beyond their own clique. The only really interesting Analytic Philosophers are those who study or take as their starting point Continental Philosophy like Dreyfus and Taylor.
Since Philosophers here in America are less educated than their Continental counterparts, and that includes in math, I think the resentment runs in the other direction. Of course, there are a few Analytical Philosophers whose work actually is about Philosophy of Math or Philosophy of Science and we presume that they know their stuff. But if there subjects of study are not math and science it is an odd thing to presume that that they know more math than their Continental Counter parts who actually know some math and science from their schooling. Just look at the ranking of kids with respect to math and science in Europe or even Asia in relation to America. We cannot fall behind like we have for years on end and expect to keep our place as the preeminent source of education within the Western world for third world societies.
I know, says one teaparty republican to another, we can solve the problem by cutting more from education. Global corporations shall solve the problem by getting their scientists and engineers from societies that believe in education and invest in it.