Philosophy discussion
Science and Mathematics
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What is Mathematics?
I agree that we have no real disagreement on the first point. For me, its interesting that mathematicians seem to bow to the idea of having a foundation for their field, but that there is also nothing like agreement about any hierarchy for those foundations.Also, I agree that it can be interesting to show equivalence in axiom systems. As for example, with the idea of computability.
It's been a while, but I thought that the idea of deriving all of math from first order logic was more than an imagined goal. I thought that, at least somewhere, Russell had stated it as a goal. I could be wrong on that. I agree that what they did was pretty spectacular in its own right.
Glad to read your last post, Duffy, I was a bit worried that my first post was clumsy, and might have created irritation. (Similarly "I give up trying to understand this Kantian stuff" really is a statement about my limited brainpower -- I hope you weren't offended, Elena.)A problem with the question "what is mathematics?" is that there was a big shift in the 20th century in the way it was perceived. There's a neat book on this, though it's been 30 years since I looked at it, namely Kline's Mathematics: The Loss of Certainty
Probably the only thing I really understand about Kant is the following: "Unfortunately, [Kant] was not a good writer, and his works are very difficult to read." Jill Vance Buroker, Kant's "Critique of Pure Reason": An Introduction (Cambridge: University Press, 1996), Kindle ed., chap. 1, loc. 32-33. However, I think it was with regard to this precise point (the subjectivity of knowledge) that Ayn Rand once wrote: "Kant was the first hippie in history." I'm quoting from memory, not having read Rand since the 1970s, and I don't recall the exact writing in which Rand said this. Additionally, I don't agree with Rand's politics or economics, but this statement struck me at the time, which is why I remember it. Do we know what theoretical physicists (for example, Lawrence Krauss) say about Kant's view?
"This does not question reality. Merely, our method of describing it."That is really the point. What is really rather ironic is that realism in the last analysis turns out to be simply a dogmatic adherence to one's method of inquiry and to the picture of reality it produces. The picture is reified and projected onto the cosmos, and we forget our own contribution to the painting of the picture. Ironically, that brand of realism undermines our quest for reality, being uncritical and unaware of our active cognitive shaping/distorting. Critical philosophy a la Kant is simply a safeguard against the pitfalls to which non-self-reflexive "realism" can lead to.
Martin, I don't take offense, and honestly, I can't blame you. Unless you feel Kant answers questions that already deeply preoccupy you, the text will fall flat. Don't you think though that Godel's Incompleteness Theorem, as well as the proliferation of irreducibly distinct systems of logic within modern mathematics, have proven all philosophical dreams of establishing a unitary foundation for mathematics to be unattainable? And if it is impossible to have such a unified foundation even within mathematics itself, then how can one sustain that whole pipedream of a universal, mathematical Logos? Or does the universe itself express itself through a prism of seemingly irreconcilable systems of axioms?
If mathematics and logic are merely the most rigorous language used to formalize the rules of human thought, and both appear to be a crazy jungle of irreconcilable systems, then reason itself seems to no longer be the classically ordered thing we imagined it to be. It seems broken into irreconcilable systems, each with its own axioms and formal rules. If we can't even unify the map of reason (math, logic), then how can we even take the leap the realists are itching to take, and speak of this crazy patchwork as an adequate reflection of "the language" or "laws" of a unitary reality? Or does the universe express itself through a crazy patchwork of distinct logical systems, too?
Alan, I am not sure what theoretical physicists would say about Kant. Physicists tend to be realists, and realists in general seem threatened by his work, despite the fact that it gels so well with findings in quantum physics (which state that one cannot coherently formulate laws of nature without reference to the position of the observer - a Kantian principle). Heisenberg I believe admired Kant; he certainly restated many Kantian principles. Here's one:
“What we observe is not nature itself, but nature exposed to our method of questioning.”
"This does not question reality. Merely, our method of describing it."That is really the point. What is really rather ironic is that realism in the last analysis turns out to be simply a dogmatic adherence to one's method of inquiry and to the picture of reality it produces. The picture is reified and projected onto the cosmos, and we forget our own contribution to the painting of the picture. Ironically, that brand of realism undermines our quest for reality, being uncritical and unaware of our active cognitive shaping/distorting. Critical philosophy a la Kant is simply a safeguard against the pitfalls to which non-self-reflexive "realism" can lead to.
Martin, I don't take offense, and honestly, I can't blame you. Unless you feel Kant answers questions that already deeply preoccupy you, the text will fall flat. Don't you think though that Godel's Incompleteness Theorem, as well as the proliferation of irreducibly distinct systems of logic within modern mathematics, have proven all philosophical dreams of establishing a unitary foundation for mathematics to be unattainable? And if it is impossible to have such a unified foundation even within mathematics itself, then how can one sustain that whole pipedream of a universal, mathematical Logos? Or does the universe itself express itself through a prism of seemingly irreconcilable systems of axioms?
If mathematics and logic are merely the most rigorous language used to formalize the rules of human thought, and both appear to be a crazy jungle of irreconcilable systems, then reason itself seems to not be the classically ordered thing we imagined it to be. It seems broken into irreconcilable systems, each with its own axioms and formal rules. If we can't even unify the map of reason (math, logic), then how can we even take the leap the realists are itching to take, and speak of this crazy patchwork as an adequate reflection of a unitary reality? Or does the universe express itself through a crazy patchwork of distinct logical systems, too? Developments in 20th century logic and math do seem to threaten the whole traditional notion of a universal Logos to which mathematics and theories of reason in general have tried to cling to all along. Regardless, many physicists like Feynman seem to remain closet Platonists in their belief in "discovering" the "laws" pf nature.
Alan, I am not sure what theoretical physicists would say about Kant. Physicists tend to be realists, and realists in general seem threatened by his work, despite the fact that it gels so well with findings in quantum physics (which state that one cannot coherently formulate laws of nature without reference to the position of the observer - a Kantian principle). Heisenberg I believe admired Kant; he certainly restated many Kantian principles. Here's one:
“What we observe is not nature itself, but nature exposed to our method of questioning.”
Elena, I do not know Kant (or modern philosophy after Kant) very well, but it seems to me what you are describing is the difference between philosophy and dogmatism. Socrates said (somewhere in Plato's Apology of Socrates or perhaps in Xenophon), “What I do not know, I do not think I know.” This has always seemed to me to be the essence of the philosophical quest. Plato distinguished between the thing itself ("reality" if you will) and the name, definition, image, and knowledge regarding the thing. Seventh Letter 342. Plato expressly stated that we should not confuse our knowledge of a thing with the thing itself. Ibid. (never mind the exoteric mysticism). Leo Strauss famously observed that "dogmatism—or the inclination 'to identify the goal of our thinking with the point at which we have become tired of thinking'—is so natural to man that it is not likely to be a preserve of the past." Natural Right and History (Chicago: University of Chicago Press, 1953), 22 (citing Lessing's January 9, 1771 letter to Mendelssohn). Strauss also said somewhere that one ceases to be a philosopher the moment one transitions from inquiry to advocacy (not his exact words but the effectual meaning of his words). By the way, Strauss was, I believe, a neo-Kantian when young. The point is that anyone can be dogmatic, but that does not mean that our attempts to describe reality are hopelessly and forever subjective. Does not this view collapse (or become vulgarized) into Protagoras's relativistic “Man is the measure of all things”? See Plato, Theatetus 151E-152E, 160C-179E; Plato, Cratylus 385E-386C; Plato, Laws 716C; Aristotle, Metaphysics 1053a35-b3, 1062b12-20.I saw a presentation by Lawrence Krauss and Richard Dawkins at Ohio State University on April 7, 2014. Although these "New Atheists" can be a bit dogmatic on the subject of religion (or at least in the way they express their views on same), they are, of course, well aware of such unexplained phenomena as quantum mechanics, the origins of the Big Bang, etc. They have somewhat more confidence than I that science will eventually solve such ultimate mysteries. Of course, Krauss's assertion that he has proved, on the basis of physics alone, that something can be created out of nothing really begs the question, because he doesn't really start with nothing. So, yes, scientists, like all of us, can sometimes be a bit dogmatic, but I do think that theoretical physics and evolutionary biology have made substantial progress in identifying the nature of reality, unless one assumes we live in some kind of Cartesian dream world or hold, with the fundamentalists, that Satan has misled us with fake fossils, etc.
OK, I'm being a bit tongue-in-cheek here, and I appreciate your elaboration of Kant, which makes a lot more sense to me than Kant himself (at least the extent to which I've waded into his writings, and my German is not sufficiently advanced to be able to read him in his own language).
Yes..reading Kant in German combs out some tangles.Duffy, yes..you are quite right, i was going to mention curved space in my 3-4-5 triangle example, but thought it an uneeded complication.
These brute facts remain..
beautiful equations expressing geometry,
speed, and position..
Knowing these variables and the mathematical dynamics of nature in motion are the beginning of "reconstructing" reality (to ridiculously inhuman precision)
Is Math REALLY just a human "invention"?
Perhaps..but i think not
man is Natures' invention....
so it follows..
his will be Hers
Am enjoying reading this..funny what Rand said...would love to hear her expand on that.
i have other things to say about Math..
and its' "realism"..mostly controversial, and some think insane
What do y'all think about the Akashic library?
Terence McKennas' DMT experiences (beings composed of pure language?
and the possibility of genetic transmission of memes?
(and, related, genetic inscription of numbers)?
"Don't you think though that Godel's Incompleteness Theorem, as well as the proliferation of irreducibly distinct systems of logic within modern mathematics, have proven all philosophical dreams of establishing a unitary foundation for mathematics to be unattainable?" (Elena.)Duffy of course gets it, but I wonder if it might not be worth explaining something about Godel's Theorem, at least for the other 2000 plus members of this group!
Russell and Whitehead built up PM (Principia Mathematica) as a foundation for maths by starting with a few very simple logical "templates", and carefully proving theorems and adding new concepts and definitions, as they went along. The system is terribly formal in presentation: please view a sample page I've just uploaded,
https://www.goodreads.com/photo/group...
A symbol like this, "|-", which you see after *56.17 and so on, means, "this is a theorem", in other words, it's "true". These are part of a potentially infinite number of possible "sentences", some theorems, some not, built up out of the same symbolism with its grammar rules. Let's call these sentences propositions. Godel explains a scheme for numbering all these propositions, 1, 2, 3, 4 and so on. This is called a Godel numbering. Godel constructs a proposition, the import of which is really this,
Proposition n: "proposition n is false"
where n is some number. If proposition n is true, then "proposition n is false" and it's false; if proposition n is false, then "proposition n is false" is false, and it's true. So either the proposition is neither true nor false (has no truth-value) or it's true and false. Either maths is incomplete (certain statements are neither true nor false), or inconsistent (both true and false).
The theorem applies to any system capable of underpinning the whole of maths, not just PM, but PM is important in providing a real system to which the theorem is applicable.
The theorem is quite fundamental, but the question is, how much does it matter? A similar result appears in computation theory. You assume you can build a Turing machine to evaluate any integer to integer function. You establish a Godel numbering of these Turing machines. Suppose the n-th Turing machine evaluates a function fn. Then define a new function F(n) = fn(n) + 1 for each n. F then differs from every fn. So F is a well-defined function that cannot be evaluated by any Turing machine. Needless to say this does not give computer scientists any sleepless nights! In the case of maths, its incompleteness is just part of its essential nature, and Godel's theorem is a lovely result.
PM spawned further work, hence "distinct systems", but that is in the nature of maths. Maths, like science, goes on for ever, just as you know that the classroom debates in Lakatos' Proofs and Refutations will go on for ever.
Gun wrote: "Am enjoying reading this..funny what Rand said...would love to hear her expand on that.
i have other things to say about Math..
and its' "realism"..mostly controversial, and some think insane
Wh..."
[I corrected a minor typo in the following on April 21, 2014.]
I find the various comments about mathematics and other arcane lore interesting but far beyond my expertise or even comprehension.
With regard to Rand, she was nothing if not dogmatic, and I have no time for her many political and economic epigones. But she had some insights, albeit dogmatically expressed, in ethics and epistemology. Moreover, she had a way with words. In checking my library, I found the Rand quote in my earlier post. (It turns out, in inspecting my library, that I was still reading Rand in the early 1980s.) Here is the quote in context (those who were not yet alive and culturally conscious during the late 1960s may not get all the witty references):
"[T]he greatest Dionysian in history was a shriveled little 'square,' well past thirty, who never drank or smoke pot, who took a daily walk with such precise, monotonous regularity that the townspeople set their clocks by him; his name was Immanuael Kant.
"Kant was the first hippie in history."
Ayn Rand, The New Left: The Anti-Industrial Revolution (New York: Signet, 1971), 64-65. See what I mean about her writing style? Rand elaborated her analysis of Kant in For the New Intellectual (New York: Signet, 1961), 30-33. She started the latter discussion with the statement, "The man who . . . closed the door of philosophy to reason was Immanuel Kant." Ibid., 30. She went on to attack Kantian epistemology and ethics. Ibid., 31-33. I cannot adequately reproduce Rand's analysis here. The book is available in paperback on Amazon and many bookstores.
Rand earlier published her sole treatise on epistemology as such: Introduction to Objectivist Epistemology (New York: The Objectivist, Inc., 1966). This is more formal than her polemical works, though still stated in absolute, dogmatic terms. It is an interesting approach, whether one agrees with it or not. I read it cover to cover in about 1982. An Expanded Second Edition (with transcripts of epistemology workshops conducted by Rand) was published by Rand's Executor, Leonard Peikoff, in 1990 (Meridian). I have not read the Second Edition.
Me again, still answering Elena.Yes, there is this idea that maths is the essential instrument for describing the Universe, and is it really up to the job? Maths, seen in this way, runs through this whole thread. This is where I really see it differently. The Universe is a big thing. Maths can be helpful in describing parts of it. It does well in modelling the behaviour of small things (particle physics) and big things (black holes and rotating galaxies). But for the in-between area where we live it is in fact not very useful at all. Today for me was a typical human day of routine events to go into the diary: took daughter to railway station; had to go back to the house for her mobile phone; then a form to complete, requiring witness's signature etc ... We spend our whole lives in this space, and it is too complex to model, mathematically or anyhow else. In fact I believe Physics is defined by its subject matter being amenable to mathematical modelling. If signing forms could be explained by a maths model as exactly as the orbit of Mercury can, it would be regarded as a part of Physics. Nor is this failure of in-between modelling due to the mere intrusion of human behaviour. If a glass of wine is blown over by the wind, no maths will predict the shape the spillage makes on the ground. And in science generally, maths plays a much smaller part than we usually admit: think of the works of Darwin.
I'm delighted that you brought up Lakatos' Proofs and Refutations. It's probably my favorite book on how math is done. For the group, its a dialogue between a teacher and some students, where they try to develop Euler's theorem: that the number of Vertices plus the number of Faces of a polyhedra (cube, pyramid, etc...) is equal to the number of Edges plus 2. I also enjoyed your summary of Godel's Incompleteness Proof. If anyone here is interested in this topic, but doesn't want to go deeply into the math, I recommend "What Is the Name of this Book?" by Smullyan. Instead of doing the math, he explores the nature of paradox in general, and there is lots of fun stuff about islands with established knights and unestablished knaves.
The Godel numbering system is incredibly ingenious. One of things that I like about it is that, at first blush, it seems very elegant. Every proposition has a number. It's sort of like a card catalogue system for Borges' Library of Babel. But it turns out that, because of the prime factorization method he used, Godel numbers are actually longer than the statements they represent. Running with my analogy, I like to think that Borge's would have liked this cataloguing system, because the catalogue would, almost necessarily, be larger than the library itself.
Martin, I agree with your final post, and it is with those thoughts in mind that I asked my questions regarding the import of the Incompleteness Theorem, as well as the emergence of distinct systems of logic. I don’t know about other people, but when I came to the study of philosophy and logic, I was quite unsettled when I discovered that there was no such thing as Logic, but instead, all we had were logicS. A bit of philosophical background is important to understand this whole issue of foundations. Crucial to the traditional founding of human reason, logic, natural science and mathematics alike was postulating a unitary, rational structure in the universe, or what was called Logos (a word etymologically connected to “logic,” “language,” “reason”). The idea was that the manifold of form observable in the universe was ultimately reducible to one simple, unifying rational principle (this is the core concept shared by our modern belief in discoverable “laws” of nature and in there being a mathematical “language” “in” the cosmos). Note how this ontological view projects our cognitive bias for simplification and selective abstraction; we cannot comprehend except through selective abstracting, schematizing, and reducing experience to a linear pattern, and so we make of this cognitive limitation an ontological principle that states that the universe in itself MUST reflect our cognitive bias and be elaborated from ultimate rational principles(!) Without this ontological postulate, this assumed essential likeness between the structure of the mind and the structure of the universe, reason, logic, maths, etc, stand on much shakier, cognitive grounds and become justified pragmatically -only- (to the extent that they are verifiable). This means you can forget about the whole talk of uncovering “laws” or the universal language – pragmatic justification doesn’t cover that stuff. But it doesn’t necessarily mean we are caught in some watertight subjective bubble, either. What emerges in experience is the result of a much more ambiguous give-and-take than either extreme suggests.
To make a long intellectual history short, the ontological postulate (Logos) was shattered by three developments: Kant’s critical turn, the existence of irreconcilable logics, and the Incompleteness Theorem (because it prevents our entertaining even in theory the idea that mathematics could ever acquire the unitary foundations that would be consonant with a unitary Logos). Setting aside the fact that modern physics seems to not reveal a very Logos-like universe, the finest irony in this situation is that it turned out that even -our- reason no longer exhibits a unitary, monolithic structure! The consensus in modern logic seems to be that the axioms that found the various systems of mathematics can no longer be treated as absolute, self-evident truths as they were in traditional mathematics. Instead, they are more akin to postulates, and they are not postulates that are shared by all the disciplines. Mathematical truth becomes relative to distinct axiomatic systems.
And this takes me to Gun’s point: I would say to you, Gun, that since we can no longer even speak of a unitary Logic (and mathematics), how can we speak of this logic as somehow a reflection of a supposedly unitary universal mathematics?
Either the splintered mathematical logics are a function of our limited, human rationality which ties itself into knots trying to grasp the order of the whole and finally cracks along the seams, or the universe itself appears splintered to us and has an order that far surpasses our rational capacity. Or, perhaps, both. In either case, all we can say about -our- mathematics is that it is ours.
But then again, I could be totally wrong on everything. I am approaching the issue from the philosophical side of things by trying to consider mathematics as one offshoot of human rationality in general, and trying to gain a sort of overview of what its limitations might be from the POV of philosophical reason. That being said, as no genius mathematician has so far solved the problem of foundations, it'd stand to reason that at least a core part of the effort must be philosophical rather than purely mathematical (though efforts like Godel's show that a formal system can reveal its own limits as well). Some rational (philosophical) interpretation seems needed as well though.
When people think of math as modeling the universe, the classic example is to compare objects in the universe to balls on a billiard table. When they hit each other, we like to imagine that their destination is perfectly determined by the first collision. And maybe so, but its also pretty astounding how quickly this type of problem gets complicated beyond anyone's ability to figure it out.In reality, its impossible to predict the outcome of a break of balls on a pool table. And the impossibility is deeply built into the problem. The following explanation comes from Taleb's "The Black Swan":
"If you know a set of basic parameters concerning the ball at rest, can computer the resistance of the table (quite elementary), and can gauge the strength of the impact, then it is rather easy to predict what would happen at the first hit. The second impact becomes more complicated, but possible; and more precision is called for. The problem is that to correctly computer the ninth impact, you need to take account the gravitational pull of someone standing next to the table (modestly, Berry’s computations use a weight of less than 150 pounds). And to compute the fifty-sixth impact, every single elementary particle in the universe needs to be present in your assumptions! An electron at the edge of the universe, separated from us by 10 billion light-years, must figure in the calculations, since it exerts a meaningful effect on the outcome. (p. 178)"
What this means is that, as a practical matter, to determine precisely the outcome of a break of balls on a pool table, you would have to know precisely the position of every proton, neutron and electron in the observable universe. In turn, that means that a computer anywhere in the universe that was calculating the problem would also effect the outcome by making the calculation. Here, the issue epistemic and not ontological. Its still perfectly possible, in some sense, that the outcome was predetermined, and yet it is also unknowable, so by any practical measure, it is random.
The pragmatic, epistemic limit is undoubtedly a serious challenge, but one that doesn't seem to stop a realist. So what if the hypothesis of a mathematical cosmos is unverifiable? One can still entertain it as ontological dogma. I was actually pursuing this same debate on another thread (see posts 7, 8 and below): https://www.goodreads.com/topic/show/...
I think its a bit unseemly for a person to insist dogmatically that something is true empirically but unknowable. It's bad enough when people make dogmatic claims about metaphysical and/or religious ideas. But if empirical statements are also, in principle, unknowable, I think that presents a major problem.
Fascinating posts above, but too busy right now to add anything except thanks for Duffy's recommendation of Smullyan's book, which I've placed on order. Here are links to the two books:Smullyan, What Is the Name of This Book?
Imre Lakatos, Proofs and Refutations
Mathematics is a tool invented by us humans to be able to preform certain tasks and use to explain something in a language that this one that I'm writing in or another can not.
I was thinking of a question concerning the nature of mathematics as a whole, and I apologize in advance if it is naive, but I cannot quite see this clearly. Given any certain field of mathematics with rules and operators, is the derivation of any analytic consequence of properly using these either a posteriori or a priori?
On one hand, it seems to me to me that, (assuming that one is making a synthetic judgment by which I take it to mean that the predicate is not contained in the subject,) we derive our judgments according to how we have derived previous ones, which is to say, a posteriori. These are examples of how we might set up the rules and operators to perform in some consistent manner.
The difficulty comes in the very way in which we use these rules and operators. Anyone who has struggled with a mathematical proof has taken various logical paths (which did not yield the desired result,) before arriving a sequence of order in which certain operators, rules and theorems might be used to provide proof thereof.
On the other hand, assuming that we are making the same synthetic judgment or judgments using the same given rules and operators, how is it that we are able to make any judgment of truth or falsity to begin with? If a predicate is not contained in a subject, logically, it seems that, for better or worse, we are making an apriori judgment. Thus it seems probable that at least some synthetic judgment of truth or falsity in a given system, must be a priori. If this is true, then it would seem that the floodgates open and allow that any synthetic mathematical judgment may be apriori.
Rhonda, I'm not sure I'm following your argument exactly, especially toward the end. This sentence threw me: "If a predicate is not contained in a subject, logically, it seems that, for better or worse, we are making an apriori judgment." In the empiricist view of the distinction, where a predicate is not contained in the subject, as you put it, the proposition is a posteriori.The distinctions you are using sound like they come from either the empiricists or from Kant. Before Kant, it was supposed that there were apriori analytic statements, and a posteriori synthetic statements. The major breakthrough of the Critique of Pure Reason was to show that there are synthetic a priori propositions, notably ones dealing with space and time.
I don't know if this is related to your concern or not, but it sounds like it could be. The reason for my doubt is because I'm not sure that I accept your explanation of what makes something a priori.
Duffy wrote: "Rhonda, I'm not sure I'm following your argument exactly, especially toward the end. This sentence threw me: "If a predicate is not contained in a subject, logically, it seems that, for better or ..."The question I am posing refers only to systems of mathematics (which may or may not have any reference to some empirical reference.) For lack of confusion, I suggest that we consider only mathematics which we cannot apply. While one may argue that all mathematics will at some point be applied mathematics, I think this puts the cart before the horse; regardless, this seems to me a separate argument.
Hence my statement would not apply to statements of the type of "Three oranges remain on the tree," but would apply to any derivation of a synthetic truth of a mathematical nature. In essence, I am posing the question as to whether mathematical statements which appear to me non-empirical, (which is to say, non-applied logical predications,) ought to be considered apriori.
As I understand it, a posteriori knowledge comes from experience of the world. A priori knowledge comes from the application of reason alone. So, yes, purely derived math propositions, for example that the square root of two is not rational, are a priori. Whether a derived proposition is analytic or synthetic is a different question.
Kant thought of them (or many of them) as synthetic apparently,"He argues that even so elementary an example in arithmetic as 7+5=12, is synthetic, since the concept of 12 is not contained in the concepts of 7, 5, or +."
From the Stanford Encyclopedia,
http://plato.stanford.edu/entries/ana...
But that is hardly the view today, surely? The propositions of maths are, or are supposed to be, analytic. Russell's definition again,
"Pure Mathematics is the class of all propositions of the form "p implies q," where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants."
By logical constants, Russell means words like "and", "or", "not", "implies", ... and the variables repeating in p and q means that you don't need to go beyond the premiss p (looking for empirical evidence) to get the truth of q. In other words, it's analytic.
But Rhonda, I don't really understand your two posts, and doubt if what I've said helps answer them. Could you perhaps illustrate your point with some example?
"But that is hardly the view today, surely?"It is still the view of intuitionists. Russell's formalists, you will recall, waged quite a hot battle against Brouwer's intuitionists. Ironically, formalists lost ground after Gödel's incompleteness theorems were introduced, while intuitionism still remains as a kind of illicit alternative view of the foundations of mathematics.
I think though that although it is "hardly the view today," Kantian intuitionism (and the view that mathematical truths are synthetic a priori, and not analytic in nature) is the more rational view. Formalism has lost its philosophical ground during the 20th century; it is only still "tenable" if you basically bury your head in the sand and choose to proceed in the direction of formalization in spite of the arguments. Sort of like you mentioned earlier, Martin - Godel's refutations of formalist foundationalism didn't stop anybody; they still went along with the quest even though it was demonstrated from within mathematics to be fruitless in the long run (fruitless because it cannot be completed, even in theory, and the quest for rational foundations becomes endlessly postponed).
Well, the ideas of Brouwer and the intuitionists seem to me so strange, and to create so many problems, that I wonder if the followers of Brouwer did not take them into realms which he never intended. There is so much one could say here, but let me give one example of the sort of thing I mean. According to the intuitionists' thinking, theorems become true at the moment of proof, so to assume a theorem false in order to get its truth (reductio ad absurdum) is rejected by them, because prior to proof the theorem is neither true nor false -- it is in a sort of "grey" state. So they reject Cantor's proof that the real numbers are uncountable (see post 50), because it relies on a classic reductio ad absurdum argument. Besides (or perhaps as a consequence of this), they don't believe in these "big" infinities anyway. All this is strange enough, but look at the actual proof, which is quite easy really,
http://www.math.hmc.edu/funfacts/ffil...
Now note: on the right is a matrix of digits. The counting of the reals goes down, each real goes across. The constructed number is one that differs at each decimal point from the green number in the diagonal, 3.43625... For this reason it's called a diagonalisation argument. But diagonalisation arguments like this turn up elsewhere in maths, for example, in constructing function F(n) of post 66 above.
And (and here's the killer) it is also used in the proof of Godel's theorem. So by the intuitionists own standards, Godel's theorem, the very thing which is supposed to overturn the formalists position, is invalid.
My own suspicion is that Kant thought of mathematical truths as synthetic because of the influence of geometry. Our understanding of space is not analytic, geometry is to be understood as a description of space, so geometric truths are synthetic. The idea of space failing to be Euclidean, or geometries describing other types of space (Duffy's example of triangles in spherical geometry is very appropriate here), was quite remote from 18th century thinking.
What about something like Alain Badiou's thesis that "mathematics is ontology?" That is as a "non-representational but schematic ontology...that does not claim to re-present or express being as an external substantiality or chaos, but rather to unfold being as it inscribes it."(Badiou, Being and Event.) This ontology, wedded to the field of set theory, is "a performative but non-specular unfolding of being." This is so, the argument goes, because set theory does enact what it speaks of (the ancient philosophical fantasy)but not in a self-reflexive way. Since Godel's Incompleteness we know that a theory cannot prove its own consistency, and furthermore set theory cannot represent itself in its totality since the set of all sets is non-existent.
What such a non-reprentational ontology can do is present inconsistent multiplicity and the structure of non-ontological situations.
I realize this sounds a lot like Elena's comment about ontological dogma. Badiou, however, wants to reverse the Kantian question and ask "pure mathematics being the science of being, how is a subject possible?" rather than positing the the transcendental subject as the grounding possibility of pure mathematics.
What Badiou believes to have found is a way to dispense with mathematical objects and to have mathematics present nothing but presentation itself. In the end, I suppose, mathematics(ontology)functions as a metalanguage. It is not that being is mathematical, but that the science of being qua being is expressed mathematically.
Joshua, Badiou's take is very interesting. However, might not one steer Badiou's claim away from the pitfalls of ontological dogmatism by revising it as "mathematics is as close to ontology as we can get?" I can accept that, plus the performative, non-representational view. If one sees it this way, then there is ultimately no conflict between Kant and Badiou. One also keeps the critical, epistemological question first, thereby avoiding ontological dogmatism of all sorts. Martin, I should clarify. I suggested that Kant could be seen as a kind of forefather of intuitionism (and epistemological constructivism), but Brouwer's and Kant's positions are not interchangeable. I am not sure to what extent your refutation of Brouwer's stance affects Kant's critical position as the two lived in quite different conceptual universes. I also think that if one reads the First Critique, one can filter out what has been refuted by further developments in logic, in geometry, in mathematics, and in natural science (all of which Kant believed had already found their ultimate paradigmatic grounding in his time). However, the basic arguments for his position as outlined in the Critique are still sound, as they are concerned with the rational (transcendental, in his terms) foundations of mathematics. The basic notion that mathematics (not just geometry) is constructed by the mind on the basis of a fundamental intuition of space (which is another way of saying that mathematical truth is synthetic a priori), is not refuted by the introduction of non-Euclidean geometries. To me, it makes more sense that Russell's view of mathematical truth as analytic and somehow discoverable in some quasi-Platonic logical atomist realm of forms. It makes sense because it deals with mathematical truth as an object of the mind, while views similar to Russell's still must struggle against the same difficulties that beset traditional representational theories of knowledge (ie, figuring out how the mind can access the quasi-Platonic realm of mathematical truth, as well as what kind of reality this realm was supposed to have).
Sometimes I feel really stupid. Here, in a discussion on ontology, I got stumped by the statement "the set of all sets is non-existent." In just about any context I would have trouble with this statement, but particularly so in a post which is making a point about math as ontology.So a couple of questions. In what sense does any set "exist"? How about the null set? If a set of sets can exist, why not the set of all sets? Wouldn't it make a difference by what we allowed to be a set in the first place? There may be something obvious here that I'm missing, but at the moment I'm definitely puzzled.
Duffy, The statement that the set of all sets does not exist(which is taken from Badiou) refers,I think, to Russell's paradox. In naive set theory a set is any definable collection. Russell proposed that a set of all sets that are not members of themselves leads to a contradiction since such a set would both be and not be a member of itself. It is my understanding that it is on the basis of this paradox, or some variation of it, that the universal set is generally regarded as inconsistent.
I'm aware of Russell's paradox. Another possible problem with a Universal set is that it would violate Cantor's theorem that a power set has a higher cardinality than the set itself. The result of this is that, to develop a consistent set theory, you either have to reject the Universal set, categorize it as something other than a set, or reject the Axiom of Comprehension. All three approaches have been done with some success and lead to different set theories.What does any of that have to do with whether the Universal set "exists"? A set theory that uses the axiom of comprehension will disallow the use of a universal set. A set theory that restricts comprehension and thus avoids the Russell paradox can also allow a universal set. If the same set theory sticks to intuitionist principles, it will also not be bothered by the inconsistency with Cantor's theorem.
Also, it's possible that Wittgenstein's approach around section 3.33 of the Tractatus resolved these problems, though its worth noting that both the followers of Russell and Wittgenstein seem to have ignored his solution. As I understand it, he basically rejects any form of self reference as being allowed in his system, and I believe this restricts the axiom of comprehension, avoids the paradox, and eliminates the need for a theory of types. I haven't read Quine's New Foundations, but I understand that he also developed a working set theory that restricts the axiom of comprehension.
I remember a mathematician once suggesting to me that problems arise when sets get too big. The set of all sets is really too big, according to his thinking. After all, what does it not contain? In the compost bin at the top of the garden I have a mass of brown worms eating the decaying vegetable matter. They form a set, and so get a place in the set of all sets, as do "the stars that shine and twinkle on the milky way", and anything else you can think of. The set of all sets not members of themselves is almost equally gross, since most sets are not members of themselves. Perhaps this lies behind Badiou's thinking: wanting to bar this monster (as Lakatos would say).And this word "existence" seems to create many puzzles in trying to decide what maths is. An earler poster said maths doesn't really exist outside the mind of mathematicians. But that is little different from saying archaeology doesn't really exist outside the minds of archaeologists. I think it comes up because so much of maths is made up of "there exists" statements. ("There exists a point inside a triangle equidistant from the three sides", etc etc)
Logically, there is a case for not saying "X exists" when X is a proper name. (I think Russell takes this line in An enquiry into meaning and truth.) This seems odd at first, but "Hamlet exists" is true because Hamlet is a character in a play, and is quite different from "Hamlet is a real person." If a name refers to nothing and you say "Lameth exists", the response would be "What is Lameth?" In a sense, proper names presuppose some form of existence. Proper names take predicates ("Hamlet is Danish") and existence is not a predicate. Existence goes with descriptions. This is especially true in maths:
"There exists numbers which can be represented as the sum of two non-zero cubes in two different ways"
1729 = 10cubed + 9cubed = 12cubed + 1cubed etc
Here, we assert existence of described numbers, but
"Does Z exist?", where Z is the system of integers, is not really a proper question, at least not for the mathematician. Z is really like Hamlet, a proper name. Similarly the empty set, and so on.
First off apologies to Duffy. I rather suspected you were much more versed in mathematics than I, and of course you have confirmed this. I will have another go at answering your question, nonetheless. What Badiou believes to have found in set theory is the mathematical theory of the pure multiple. For Badiou ontology has always foundered on the question of the one and the multiple. He writes: "Since its Parmenidean organization, ontology has built the portico of its ruined temple out of the following experience: what presents itself is essentially multiple; what presents itself is essentially one."(Being and Event,23) If being is one then the multiple is not. If what is presented is multiple then it would seem that access to being occurs outside of presentation, an unacceptable conclusion. The conclusion that the multiple is is equally problematic because what is presented has to be counted as one.
For Badiou the answer to this dilemma is to reject the thesis of a being of the One, while maintaining an operative principle of counting. "The one, which is not,solely exists as an operation." The multiple is the regime of presentation, and the one its operational result.
Now, back to set theory. Badiou defines a set as "what counts-as-one a formula's multiple of validation." Here is an interesting, though rather longish quote:
"It is quite remarkable that, in the very moment of creating the mathematical theory of the pure multiple -termed 'set theory' Cantor thought it possible to 'define' the abstract notion of set in the following famous philosopheme: 'By set what is understood is the grouping into a totality of quite distinct objects of our intuition or our thought.' Without exaggeration, Cantor assembles in this definition every single concept whose decomposition is brought about by set theory: the concept of totality, of the object,of distinction, and that of intuition. What makes up a set is not a totalization, nor are its elements objects, nor may distinctions be made in some infinite collections of sets (without a special axiom), nor can one possess the slightest intuition of each supposed element of a modestly 'large' set.'Thought' alone is adequate to the task, such that what remains of the Cantorian 'definition' basically takes us back -inasmuch as under the name of set it is a matter of being- to Parmenides' aphorism: "The same, itself, is both thinking and being.'(B&E,38)
He goes on to suggest that the profound significance of the paradoxes from which set theory emerged, of which Russell's paradox is the most well known, is that control of language does not equal control of the multiple -"the multiple does not allow its being to be prescribed from the standpoint of language alone." Or again - "Language cannot induce existence,solely a split within existence." Any cuts and compositions made within language presuppose an initial presentation of (multiple)Being that language does not guarantee. The existential guarantee, for Badiou, is the null set because it presents literally nothing and therefore functions as a ground for presentation as such. Martin's comments about proper names are a helpful example of how this works.
Joshua, thanks for the detailed reply. I don't have that much trouble with the math, but when it comes to wrapping my head around the ontological theories, I tend to develop headaches. I'm going to take some time thinking about your explanation, and then I'll reply further.Martin, the solution of disallowing existence as a predicate works pretty well in formal logic. But that doesn't solve the problem unless one insists that formal logical expressions are in some sense superior to ordinary ones.
Take the sentence "Homer did not exist." I think most people would understand the sense of this statement, and might be able to give reasons why they thought it was true or false. It's possible that the statement is non-sense, but it's not obvious to me. And as for Homer, my own opinion is that he did not exist, and that the Iliad and Odyssey were written by another guy with the same name.
Your comment may be interesting, but Plato and this time i join his opinion, would not agree -- due to evident reasoning.
Elena, The question your post raises,for me, is one of praxis. This may not be the thread to address it, and I think our discussion in the technology thread got to the same place. The question,I suppose,is how there can be a communal/material epistemological practice, or are we condemned to be beautiful souls (in Hegel's sense)?
Re praxis (Joshua's post 93): Although I don't have the background to follow the abstruse mathematical discussions addressed in these posts and am accordingly not certain whether the present post is responsive to any of the earlier ones, I would venture to observe that, notwithstanding the pretensions of some schools of twentieth-century political science, mathematics does not, and cannot, address the questions of ethical and political philosophy. See Aristotle, Nicomachean Ethics 1094b. As Aristotle observed, "One would speak adequately if one were to attain the clarity that goes along with the underlying material, for precision ought not to be sought in the same way in all kinds of discourse . . . . [F]or it belongs to an educated person to look for just so much precision in each kind of discourse as the nature of the thing one is concerned with admits; for to demand demonstration from a rhetorician seems about like accepting probable conclusions from a mathematician." Ibid. Aristotle, Nicomachean Ethics, trans. and ed. Joe Sachs (Newburyport, MA: Focus Publishing, 2002), 3. (Note: Aristotle obviously did not anticipate the discipline of statistics.)
Additionally, unless and until further study and reflection convince me otherwise, I accept the following as working hypotheses: (1) "Existence exists" (Rand, following Aristotle), and (2) "I am, therefore I think" (Rand, following Aristotle and correcting Descartes). OK, go ahead and call me a naïve, unreconstructed realist (LOL)! But don't call me "dogmatic": I reserve the right to change my mind after I complete my study of modern and postmodern philosophy. Check back with me in about five years (if, given my advancing age, I still exist at that time; if not, my epitaph will read: "I am not; therefore, I think not").
I've been thinking a lot about Elena's post #85, and do now see the attraction of the idea of maths as "synthetic, a priori". In other words, being in this thread has shifted my thinking, which is quite nice, really.One thing I would stress though, is that the work of Russell/Whitehead is not quite on the same level as the other philosophers who might appear to rival them. The process throughout the 19th century was to establish the mathematics of functions and the calculus on good foundations, with exact definitions, axioms, and theorems built up one on another, like the theorems in Euclid. When taught today, it is called Mathematical Analysis. One might say the result was very successful, but in the process the certainty of mathematical truth seemed to recede from its "platonic" position. The story is told in a popular form in Kline's book. This program became all-embracing, and led into 20th century maths. Russell was part of the program, working not on the pinnacles and towers but the deep foundations. There was, I think, a sense in which what he did could not really have been done in a significantly different way. Of course, details might have been altered, but Russell was often quite relaxed about alternative methods. He had no objection to trying three-valued logic, or to learning of the computer-generation of the earlier theorems. (There is a book Dear Bertrand Russell of his letters where he says this.) Perhaps it isn't so much "Russell as formalist versus x as intuitionist" as the way Russell's work should now be understood.
Incidentally, I remember someone who knew this field well saying to me that Godel's theorem was much more challenging to Wittgenstein's ideas than it was to Russell's.
Possibly challenging to the ideas in the Tractatus, but not so much to his later views. Tractatus takes an almost Platonic position about the world as a body of propositions.
Duffy, yes! I remember now that that was his argument. And it was just the Tractatus he was talking about. (Your grasp of all this really impresses me, you know.)
Alan, In response to your posts 94 and 95; I don't see how Rand's statements can launch us into the adventure or discipline of thought. There is at least an apparent gulf between the formulations "there is existence" and "existence exists." We certainly would not want to say "existence is an existent," which, I think, is the temptation of undisciplined thought.
When Diogenes the Cynic was confronted with Zeno's paradoxes of motion he responded by taking a walk. One of Diogenes' students praised his master's wit, and Diogenes gave him a good thrashing because "a factum brutum is not a philosophical proof." The point I made earlier about mathematics and ontology was certainly not meant to suggest that ethical and political philosophy can be derived from mathematics. Nor even that ethical and political philosophy can be represented mathematically. The role offered for mathematics was that of schematics, or, to put it differently, a set of rules for describing being.
What is at stake, after all, is our method of describing reality, rather than reality itself.(Posts 57&62). The caveat is that language must relate to reality intrinsically as well extrinsically. Otherwise we would be left with "existence exists but language/thought does not" or "existence does not exist and language/thought describes nothing." My own take on it is somewhere between "existence is language" and "existence appears through language." Except that, as Elena keeps pointing out, it is all too easy to lose the quality of reflexivity in the desire to fully identify with our language about reality.
The question, for me, becomes one of conviction. At what point does reality demand of me an assent that reaches beyond the limitations of my understanding? At what point do I throw myself into the fray of being without reserve? Ontology, in the way we were speaking about it, does demand a decision. As Duffy points out different versions of set theory arise depending on whether one rejects the Universal set, categorizes it as something other than a set, or rejects the Axiom of Comprehension. A decision is required.
Maintaining a critical opposition to dogma of any kind is also a decision. It may be the right decision, but it might simply be a precaution against making a (wrong) decision. Kant was heavily influenced by Pietist spirituality, and this shows up in the way his thought is shaped by the maintenance of an interior distance from the objects of thought. I am suspicious,however, of the truthfulness of this distance in terms of the material existence of human beings and the sociality of subjects. I suppose I think the distance is somewhat artificial and does not address the question of conviction and transmission of knowledge satisfactorily. The question of praxis is bound up with the question of education, which is where the question of mathematics becomes incredibly important.
Thanks, Joshua, for your interesting post. I was speaking somewhat tongue in cheek in citing Rand, though she has a point (not necessarily in the literal sense). Since it has been over 30 years since I've read Rand's Introduction to Objectivist Epistemology and almost exactly 45 years since I've read Aristotle's Metaphysics, I'm really not prepared to debate these issues right now, especially since I'm not well read in Kant and post-Kantian philosophy. I'm preoccupied right now with historical research, but I'll get back to philosophy at some point in the not-too-distant future. As previously indicated, I am primarily interested in ethical and political philosophy. I do, provisionally at least, follow Aristotle (but not, apparently, Plato if one takes his statements in The Republic as his last word), Rand, and Strauss in assuming that one can take for granted certain terrestrial "givens" for the purposes of ethical and political philosophy, i.e., human life as we know it as somehow "real" (even if far from "ideal"). But I do find the various posts in this thread interesting, even if I don't understand the mathematics or even the post-Kantian philosophy being discussed. I will probably revisit these issues sometime later this decade and may have further thoughts at that time. In the meantime, I find solace in Socrates's dictum, "What I do not know, I do not think I know." Or Pierre-Simon, marquis de Laplace: "What we know here is little, but what we are ignorant of is immense." Keep on posting, you guys/gals! I'll probably be fairly silent for the foreseeable future, at least on this thread. Oh, and I didn't mean to imply, Joshua, that you or the other posters shared the pseudo-mathematical approach of the twentieth-century behavioral political scientists. That's just my own personal grudge. I gave up on academic political science at the precise moment at which the American Political Science Review became mainly a locus of mathematical equations.
Joshua, a question in response to your illuminating post: where does "the decision" you mention come from? What informs it? Your post is starting to sound a bit Nietzschean in its focus on an act of will that posits its objects and meanings ex nihilo. Ultimately, such an emphasis on will/conviction/decision becomes anti-rational and therefore troublesome. I agree that projecting our cognitive constructions onto the world and pretending these are representations of "the ontic order" is a fallacious procedure that should be avoided, but the Nietzschean alternative of over-emphasizing praxis and will seems to veer too far into subjectivism. My own sense is that math represents the absolute paradigm for the adequate representation of reality that would be shared by all creatures with minds such as ours, and therefore ontologically situated in relation to reality as we are. Mind is part of a biophysical continuum, and its symbolic structures, whether they be math, logic, art, myth, or whatever, represent the interchanges that constitute the experience the mind has -within- that world. I think I understand now why many have a knee-jerk distaste for the Kantian, self-reflexive starting point: they believe it constricts the real into a mind-bubble. I think if you read the Critique itself, you will see that the model of reason is far more situated and interactive than most people realize, thereby effectively deligitimizing all subsequent idealist interpretations of his work. He does specify that any act of knowledge is comprised of a formal principle, which we bring to the experience, which comes to be activated by a given intuition of the surrounding world (again, "Thoughts without content are empty, intuitions without concepts are blind.") Math is a little bit different in his scheme in that it is elaborated on the basis of a "pure" intuition of space. Where that leaves the ontological status of math is a burning question; my take is that it leaves it in a strange terrain somewhere between the psychological/cognitive and the ontological - a terrain that our intellectual culture is not really conceptually equipped to deal with right now.
I seem to be coming off as a kind of Kant fan girl on this forum, and really don't want to because I am probably not an adequate representative of his POV, nor is my thought altogether synchronous with his. But I do think 20th century philosophy of mathematics and thinking on logic has set aside the Kantian arguments to their own detriment. Part of the reason philosophy is at its current, logic-chopping impasse is because Kant's arguments have not been internalized sufficiently. If they had, we wouldn't have uncritical positivists or reductionists! Ignoring Kant and as a result regressing into non-tenable ontological positions means the status of reason becomes severely impaired as it loses itself in all kinds of intellectual dead-ends (like logical atomism, again). When the instrument of philosophy, reason, is impaired, philosophy comes to a standstill. I believe the line of thinking Kant opened up really is an antidote to this, as it puts reason on secure foundations. They are limited foundations compared to the wild wetdreams entertained by some ontological dogmatists, but at least they are secure and true to our (finite) being. Now, for math, what better way to start an inquiry into its nature than to first begin with what we actually, positively, and absolutely know about it - ie, that it is first and foremost an object of the human mind. Whatever else it may be, this we can build upon. This starting point for looking at the issue makes all the difference. This really shouldn't be so controversial, and yet it is. It almost seems offensive to suggest it.
I do agree that the Kantian POV is merely a start, and that it is a schematic, incomplete, abstract, artificial and rather threadbare account of human knowledge. The reason is that it lacks a sufficiently fleshed out phenomenology such as the one Merleau-Ponty developed that would take into account the situatedness and embodiment of the knower. Kant has the architectonic, logically ultimate outlines of the principles of rationality, but drawn out in a myopic, skeletal fashion. What's left is the phenomenology, the embodiment of thought, the embeddedness of mind in world as a being-in-the-world, the historical continuity discernible between individual minds, the cultural embeddedness, as well as the intersubjective elaboration of our thoughts. I would propose that his philosophy merely provides a sound stepping stone to further philosophical development. His genius, I think, lay in his ability to identify the absolute most secure and true-to-the-human-condition starting point for thought. The synthetic a prioricity of mathematical truth secures this aim better than rival theories, such as Russell's.
Elena, that is a dazzling demonstration of knowledge and wit (in both the archaic and modern senses of the word). Question: Is not the Nietzschean emphasis on will in this sense the progenitor of "commitment" a la Heidegger, the existentialists, and the 1960s-70s New Left/hippies (before your time, I'm sure, but I lived through the latter phenomena)? I'm not all that knowledgeable about Heidegger, though I studied Nietzsche in some depth long ago. But I recall quite distinctly the New Left emphasis on commitment, and I read a fair amount of Marcuse (the aesthetic regime) in about 1970. It is interesting how you make Kant the enemy of positivism. I really will have to give another go at Kant after I finish my present historical studies. You will all have to excuse my ignorance of post-Nietzschean twentieth-century philosophy. I never could quite get started on Russell, Whitehead, et al. or on Heidegger for that matter.
Addendum: When Joshua used the word "praxis" in post 93, I assumed in my post 94 that he was referring to the classical Greek understanding of the word, i.e., practical wisdom or practical reason. I realize that Kant wrote a critique of practical reason, but I haven't read more than a few pages of it. Elena, in post 101, appears to interpret the fourth paragraph of Joshua's post 99 as a sort of twentieth-century existentialist "commitment" in abstraction from reason. As I understand the history of twentieth-century existentialism (which could be incorrect), it was precisely such a commitment to "commitment" that led Heidegger to support National Socialism. Ironically, it also led to the New Left and the counterculture of the 1960s-70s, which preached the superiority of "commitment" over reason. At the time, I saw clearly the connection between the New Left theorists (Marcuse et al.) and Nietzsche, Heidegger, Hitler, etc. Perhaps Joshua's question "At what point do I throw myself into the fray of being without reserve?" led Elena to think that Joshua was going in that direction. I'm not sure whether or not this was Joshua's meaning (perhaps he was speaking ironically or hyperbolically), but he can speak for himself. I would add only the following. Yes, the moment when we go from philosophy to praxis (in the ancient Greek sense) is the moment when we have to take a step that is, at best, probable. The ancient Greeks, not to mention the American Founders, understood this well. See my quotation from Aristotle in post 94. The point is to do the best one can in the world of praxis. This does not mean an abandonment of reason. Rather, it means that one uses practical reason (logic, evidence, etc.) to the extent possible in the ethical and political realms. One does not have to give up being a philosopher, but one has to realize that rational certainty is rarely, if ever, possible in matters of human affairs. This does not mean that one plunges into political commitment without rational analysis or mental reservations; rationalism and a healthy skepticism are always desirable and regrettably absent from today's political ideologies. Such concerns are why I am currently focusing on historical studies rather than theoretical reason. However, as previously indicated, I will return to philosophical studies as such as soon as my present historical study is concluded.
Yup, mathematics is a way to describe the nature.Igor
Elena, Well put. Where does the decision come from,indeed? The question I was hoping to get at has something to do with the cultivation of the aptitude for rational thought, as well as how we can think the place of rationality within human reason. What seems to me to be occluded in Kant's thought is the thinking of the thinker. I disagree with Kant on the fundamental question: for me the question of how the faculty of thought is possible holds pre-eminence over the question of "what and how much the understanding and reason can know apart from all experience." I am not at all convinced that a serious philosophical engagement with biology necessitates that we assign finitude as the law of being, and therefore thinking.
This brings me to the related question of the possibility of thinking a new thought, or, to put it differently the possibility of ontological difference. The reiterations of the limitations of thought,which accompany Kant,seem to foreclose the horizons of thought. I realize that I am somewhat unclear on the distance that separates Kant and Brouwer, but sympathize here with David Hilbert's sentiments on the latter: "What Weyl and Brouwer are doing amounts in essence... to provide a foundation
for mathematics by pitching overboard whatever discomforts them and declaring an embargo.." More seriously, however, I am not certain that positing the (finite)subject as a solution to the problem of possibility or unity (whether of intuitive certainty a la Descartes or synthetic judgements a priori as for Kant) is satisfactory.
Badiou's philosophy represents an attempt to think the possibility of the event within ontology(Being and Event) and subsequently to think, or show, the singular subject as "the active bearer of the dialectical overcoming of simple materialism." Badiou describes the distance between his thought and Kant's with reference to the concept of the object as follows:
"For Kant, the object is the result of the synthetic operation of consciousness. For me, the object is the appearing of a multiple-being in a determinate world, and its concept(transcendental indexing, real atoms...)does not imply any subject. But the question is far more complicated. Why? Because the notion of the object crystallizes the ambiguities present in Kant's undertaking. In brief, it is the point of undecidability between the empirical and the transcendental, between receptivity and spontaneity and between the objective and subjective.(Logics of Worlds.)
Badiou agrees that Kant is right to say that "if I leave aside all intuition, the form of thought still remains, that is the manner of determining an object for the manifold of a possible intuition." For Badiou, however, what Kant is unable to see is that thought is the capacity to synthetically think the noumenal and the phenomenal. The concept of the object, Badiou says, "designates the point where phenomenon and noumenon are indistinguishable, the point of reciprocity between the logical and the onto-logical."
What I appreciate about this attempt is that an effort is made to locate and deploy our capacity for abstract thought and not simply to confine it to an elevated but rather precious space. Can finite beings with infinite aspirations and the capacity to think,perhaps obscurely, infinity really be circumvented with the law of finitude with respect to their being-thought?
Alan, with all due respect, I tend to read Heidegger's political commitments more in terms of a policing of being-there than a wild political passion. But the follies of National Socialism attracted all kinds of philosophical partisans; Adolf Eichmann believed himself to be acting in accordance with Kant's categorical imperative.
Joshua - Having not really read Heidegger at any length or in any depth, I have no idea what you mean, but I will study him (after I study Kant) and then get back to you if I disagree. But it won't be anytime soon, as I have other projects that either are already in the works or that I have already planned. Thank you for your interpretation, which I will keep in mind.
Philosophy
Books mentioned in this topic
What Is the Name of This Book? (other topics)Proofs and Refutations: The Logic of Mathematical Discovery (other topics)
Mathematics: The Loss of Certainty (other topics)
What Is Philosophy? (other topics)
What Is Philosophy? (other topics)
Language is not reality. Rather it is a method for describing reality. Therein lies the problem with asserting that mathematics is the "language of the universe." The universe is indifferent to math, it simply does what it does. We created math as a way to describe what the universe is doing. Therefore, being a product of the human mind, psychology applies.
This does not question reality. Merely, our method of describing it.