/lit/ - Literature

>>13551306
Greek Mathematical Thought and the Origin of Algebra

>>13551306
I kind of like this, why have I been avoiding calvin and hobbes

>>13551347

In the late 19th century and early 20th century the entire mathematical world collectively shit itself trying to ground/found mathematics in some kind of phenomenological or naively phenomenological (neo-Kentian), linguistic logicist, or Platonist-logicist basis. Hilbert, Husserl, Frege, Brouwer, Cantor and many others all went at it.

The nominalists and set theorists eventually won, and the "foundations" of math are presently "don't think too much about foundations, just do math" mixed with naive "math is math, it must have foundations, right?" mixed with a few batshit set theorists who think their mysticism is the platonic essence of math but would deny being metaphysical. So if you go back to that (unresolved) philosophical debate in its heyday you can essentially get right back into the weeds with it.

In general the question of the essence of math is really interesting because it's a bunch of questions folded into one: What is the fundamental phenomenological nature of arithmetic? Surely we're not "counting" sums like 782,928,134 in the same sense as when we "count" 5 or 6 abstract dots in our mind's eye? So for sums that can't be "checked," in such an immediate, crypto-Cartesian "clear and distinct idea" sense, are we actually building castles of symbolically interlinked propositions? But then what is the nature of a proposition? Surely it's not "self-standing," somehow correct "in itself" regardless of anyone actually presenting it phenomenally and acceding to it. Worse then, what is the status of propositions that AREN'T currently being checked, but which are assumed without even being presented as the axial conditions of some other proposition being considered?

Likewise, what is the status of apodicticity? Does A=A because "one simply must accede to that!" or does A=A because people empirically DO generally accede to it? What is the epistemological status/importance of someone saying "No, I don't think A=A."

Even worse than all this is when you get to the point where vast interlinked symbolic structures (i.e., math) are now being used to represent "objects" that aren't real. A complex arithmetical sum about discrete physical objects is possible to "verify" in at least some way -- this was the view of Morris Kline, that math has historically been linked to physics and phenomenally presentable objects, and not image-less abstractions. But now we ARE essentially deriving a metaphysics (of higher-dimensional "things" and their behaviours) from simpler arithmetical-symbolical systems. We're talking from math to reality, rather than the other way around. Bachelard writes about this in the first half of his career, especially on Einstein. He says that math is shearing away "thingly" metaphysics and presenting things in pure abstraction, but clearly physicists now speak freely about a whole invisible ontology of higher dimensions.

>>13551389
I appreciate the earnest effortposting, it's rare to see nowadays and is all that keeps this sickly corpse of a board still kicking. Do you know of any good secondary literature that presents these issues in a relatively condensed form? I'm also interested in the early historicity of mathematician-metaphysicians like Pythagoras or Plotinus but I'm aware that I only have the dimmest idea of all that as a genre of thought; I'm wand'ring uncompassed.

Pythagoreanism is actually a kind of either devaluing or elucidation of Primally old shamanic wisdom expressed fractally in almost cargo cult like fashion (compared to its unbroken form) in ancient Egyptian and ANE cultures and cosmologies etc.

Thus it follows that Platonism is a degenerate and secular form of pythagorianism and neoplatonism is a partial affirmation if not return toward the reconciliation of the profoundly illogical aspects of "philosophy". The commentary tradition of neo pythagorean, middle platonist and of the "commentators" are contrary to popular belief, not banal rote missives rather invocations of punctuated equilibrium born of repitition. This repitition guarantees to prove unequivocally that univocal agreement is practically impossible.

There is an inherent geography and corporeality to language, at it's limits (this limit is paradox/Demi-god) and when these limits are stroked and fluffed into affirmed arousal they become paradoxes which in turn become vaginas which are liminal zones, thresholds and portals in the mind of the human.

Western civilization has lost its roots in true magical training but Platonic philosophy is derived from this original mystical magical training. So Plato used the Archytas version of Pythagorean philosophy and so the "harmonic mean" did not exist in traditional or "orthodox" Pythagorean philosophy.

This is the "three-in-one" unity - you can not separate one number from the other - and it has to be contemplated in silence for a long time in a pure environment. Plato was an oil merchant and his goal was not this real mystical training. Sure he relied on it but instead turned it into a materialistic mysticism - based on using the new geometry created by Archytas from the Pythagorean philosophy. So the Pythagorean math is deceptively simple - just like Taoist yin-yang philosophy - but to practice it on a sincere level is exceedingly difficult. Meanwhile there is all this fake "sacred geometry" using irrational numbers which really is not Pythagorean philosophy at all. It's the perfect way to cover up the real philosophy - a "bait and switch" tactic. So that way you say Pythagoras - and people go - Oh I know him he created the square root of two! haha. Quite the contrary. So the myth that someone who discovered his "secret" of the square root of two and then told about it was then killed - this is the typical myth to hide the true Pythagorean teachings. The "Orthodox" Pythagorean philosophers ONLY used the Tetrad - 1:2:3:4 - they did not even use the number 8 - so they could not have even created the 9/8 ratio as the major 2nd music interval. In other words the scale was just the octaves and the perfect fifth and perfect fourth. You find the same music intervals in all human cultures - and it derives from the original human culture - the Bushmen of Africa who were the original Pythagorean-Taoists - using the 1-4-5 music intervals as alchemical training.

>>13551664
Read the CTMU

Mathematics is a logically idempotent metalinguistic identity of reality which couples the two initial ingredients of awareness: perceptual reality (the basis of physics), and cognitive-perceptual syntax, a formalization of mind. The explanation has been reduced to a few very simple, clearly explained mathematical ingredients. This paper contains no assumptions or arguable assertions, and is therefore presented as an advanced formulation of logic which has been updated for meaningful reference to the structure of reality at large. This structure, called the Cognitive-Theoretic Model of the Universe or CTMU, resolves the problems attending Cartesian dualism by replacing dualism with the mathematical property of self-duality, meaning (for reality-theoretic purposes) the quantum-level invariance of identity under permutation of objective and spatiotemporal data types. The CTMU takes the form of a global coupling or superposition of mind and physical reality in a self-dual metaphysical identity M:L< >U, which can be intrinsically developed into a logico-geometrically self-dual, ontologically self- contained language incorporating its own medium of existence and comprising its own model therein. y.

>>13551684
any good books on pythagoreanism?

>>13551694
Myth of Invariance by Ernest G. McClain

>>13551306
On a similar note, what are the good histories of mathematics? I think it's better to ground oneself in the history before tackling the philosophy, so I'm curious if anyone has any good suggestions.

>>13551684
Much less into this than the previous effortpost. Is this actual Pythagoreanism or just post hoc pseudo-Christian esoterico-alchemical archetypification, Pythagorus made out Hermes Trismegistic?

>>13551700
very thanks

>>13551687
The decline of /lit/ is the fault of strenuous drivel such as this masquerading as true effortposting

>>13551664
Thanks for calling my spergout effortposting, anon.

If you know basic math already, I'd say go read Morris Kline's big history of mathematics. It's a lot of fun. Personally the key to understanding many of these issues for me was trying to understand the ontological and epistemological status of differential calculus, especially when people would say things like "well, we couldn't 'solve' Zeno's paradoxes until infinitesimal calculus came along," which always seemed wrong to me, because they didn't SOLVE the paradoxes, they just supplied a different ontology (which assumes a priori, unlike Zeno, that a sum approaching a limit of an identity = the identity itself). Then I read Kline and it's all right there in the first few chapters.

There's a book I've been trying to find for the last five minutes called something like "The Crisis of Mathematical Foundations: 1880-1930" or something, but if you search around for general history books on the crisis you will find plenty. The IEP article on Foundationalism has a lot of recommended readings that are short articles.

Ultimately it's the primary sources and the debates between them that are the most interesting. It took me a very long time to understand how Husserl got from being what seemed like a naive platonist/aristotelian to a phenomenologist. It seemed to me that anyone "attuned to" the transcendental way of thinking, especially as much as a phenomenologist must be, could never have been a naive realist in the first place. But once you understand the intellectual context in which Husserl was doing his work on arithmetic, it makes more sense. Same for the intuitionists and even the formalists. It also shed light on Frege for me, again, someone I couldn't understand because I couldn't tell if (and if so, how) they were a naive realist.

There's a really nice article I found about Frege's and Hilbert's correspondence:
Herbert Breger, "Tacit Knowledge and Mathematical Progress," in The Growth of Mathematical Knowledge (ed. Grosholz and Breger), 221-231. It shines an interesting light on Hilbert's formalism. The best part of the article is when Frege asks Hilbert how his formal structures apply back to reality, and he simply says "with good will and tact," which to me is a very proto-Wittgensteinian phrase.

Also Wittgenstein obviously. I would skip the Tractatus until you are willing to read a lot of the intellectual background necessary to form an opinion on its meaning, because the TLP is very contested and poorly understood. But if you already study phenomenology, and especially hermeneutic phenomenology, you should have no problems whatsoever reading the Logical Investigations + On Certainty, and from there you can go easily to Remarks on the Foundations of Mathematics. The best reading of Wittgenstein IMHO is that he was, essentially, a good hermeneutic phenomenologist, but not a complete sceptic about transcendental phenomenology. So his insights into math are really valuable.

>>13551772
Also would recommend Ian Hacking on the "historical ontology" (basically phenomenology) of conceptualizing probability/chance/statistics, and Lorraine Daston/Galison on Objectivity (book)

>>13551389
So basically if we can't see it its not real?

>>13551365
I don't get it

>>13551389
Alfred North Whitehead is amazing for such good reading, and is very accessible if you start with his earlier work. This is a great overview to help you decide if you'd like to read him: https://www.iep.utm.edu/whitehed/

His philosophy is among the current of "shearing away "thingly" metaphysics and presenting things in pure abstraction" that >>13551389 describes, but rather he considers thing-ness as an abstraction, and the true concrete nature of reality is process and relationships. His work was robust enough to have an element of it formalized: https://en.wikipedia.org/wiki/Mereotopology

>>13551306
I would recommend that you just start learning math, even at it's most basic level. I imagine you already know some basic algebra, but take some time to pick up a bit of Calculus and Linear Algebra. Both of those subjects lead to the higher levels of mathematics.

I haven't personally done a lot of reading into the philosophy of mathematics, but if I were to recommend a start into mathematics, and specifically calculus and algebra for someone looking for more philosophical applications:

Infinite Powers - Stephen Strogatz; Inf Powers is the beginnings of integral and differential calculus from its beginnings with Archemidies and his contemporaries (as well his predecessors) to the more modern applications today in differential equations and real state analysis.

A book of abstract algebra - Charles C. Pinter; I haven't actually made a significant amount of progress in this one because of school, but it's more based in teaching bits of theory followed by work to solidify the ideals.

I realize this isn't exactly in the vein of what you were looking for, but if you are unfamiliar with mathematics as a whole, learning more math can't be a bad idea.

Just keep practicing and reading and you'll do well! Math is fun!

>>13551783
It's more like, taking the question of "how do we know it's real?" as a problem.

The reason I like the Hilbert phrasing so much is that it captures the fundamental problem of math: it's insanely powerful and useful to the point that it seems to "work on its own," and that it seems infinitely bigger than any one individual practicing it, yet somehow each dopey concrete individual is the one responsible for plugging it into reality, in whatever context he works.

You can come up with crystalline spires of infinite meta-recursive complexity in set theory, as harmonized by hundreds of scholars over a hundred years going "yeah this all makes sense," and at the end of the day some comparatively puny little human being has to come along and actually supply the conjunctive "THEREFORE" that connects it back to a real, concrete case (in physics, in the phenomenal manipulation of quantities or objects, etc.).

At the end of the day I'm more interested in mathematical platonism than anyone, and I want to know why math DOES seem to correspond to and predict real physical things. For example, lately I've been really interested in the leapfrogging of mathematics and physics in the early to mid 20th century as people outright predicted the existence of subatomic particles. That wasn't some purely constructivist affair, the math was "doing something" there, it WAS "corresponding" to reality. But what does that mean for reality? What if the way in which our math corresponds to reality is actually an impoverished and low-fidelity version of the way some hypothetical phenomenologically enhanced supermath could correspond to, or plug directly into, reality?

What needs to happen at this point is a careful deconstruction/destruktion and re-construction (in the sense of reconstituting the Ereignis) of mathematics-physics from the ground up.

Also forgot to mention Road to Reality by Penrose.

>>13551772
Not the OP, but I'm stealing thi post for future reference and there is nothing you can do about it.

>>13551824
Bergson?

Basically,

Timaeus by Plato

Introduction to Arithmetic by Nicomachus

Metaphysics by Aristotle

In that order.

>>13551824
>The reason I like the Hilbert phrasing so much is that it captures the fundamental problem of math: it's insanely powerful and useful to the point that it seems to "work on its own," and that it seems infinitely bigger than any one individual practicing it, yet somehow each dopey concrete individual is the one responsible for plugging it into reality, in whatever context he works.

Ths is a very interesting obsrvation. As someone who has worked in applied maths, I'm a bit incensed that people often forget how much work it takes to "use math" in practical contexts. Maths is useful also because we've taken great pains to use it.
Still as you say it has an almost coercive force of its own, reading a good math proof is sometimes being compelled to agree.

>That wasn't some purely constructivist affair, the math was "doing something" there, it WAS "corresponding" to reality. But what does that mean for reality? But what does that mean for reality? What if the way in which our math corresponds to reality is actually an impoverished and low-fidelity version of the way some hypothetical phenomenologically enhanced supermath could correspond to, or plug directly into, reality?
You've probably read it already, but Bachelard somewhere mentions that the "empirical validation" we get for those models are through hard-won statistical analysis of complex observation machines whose design and construction is beyong the power of any single individual. So even the empirical aspect of science is getting more "ghostly" in a way.

I've been pretty fascinated by your posts itt since they seem to reflect my own questioning so closely. Could you tell us more about your background, and how you came to this? Have you read the work of Lautman, or works on philosophy of mathematics by mathematicians (thinking Poincaré, Hadamard, to a lesser extent Hardy, Schwartz and Grotehndieck).

Also is there a shortlist of the main thinker and main works you consider essential in this line of inquiry? Besides those you already cited.

>>13551389
Isn't this similar to Godel's Incompleteness theorem: that we can have proofs within mathematics, but we can't have a proof for mathematics itself?
I always thought this interesting in regards to all fields that use mathematics epistemology as a useful heuristic aka "Don't think about it, just shut up and calculate". Particularly the scientific fields (especially the purely theoretical ones), which are essentially built on these metaphysical foundation of mathematical verification but will never acknowledge this and claim their fields are purely reductional and materialistic. I always found this stance to be disingenuous from the scientific community. You end up with materialist philosophers rejecting explicitly rejecting metaphysics while implicitly using metaphysics in their statements.

I'd probably expect it to be likely for a mathematician to not be an atheist than it is for a scientist to be an atheist. I don't see how one can devote their life to mathematics without questioning the epistemology and ontology of it; especially since no one has ever physically observed mathematics.

>>13551965
>that we can have proofs within mathematics, but we can't have a proof for mathematics itself?
Gödel Incompleteness Theorem is a very significant achievement but it's still much, much more specific than that. But you're right that it was pretty much a death blow to Hilbert's formalist project.

Hilbert dreamt of reformulating traditional mathematics, particularly arithmetic, into a set of elementary procedures that could be applied automatically and which would be self-sufficient and self-consistent.

In essence, that theorem tells you that Hilbert has to give up somehting: either he renounces limiting himself to elementary operations, or he has to scale back to a lesser form of arithmetic (for instance Presburger's arithmetic, which is like usual arithmetic but without multiplication), or he has to accept he won't be able to prove that his arithmetic is consistent with those same arithmetic means.

Fwiw you can prove traditional arithmetic is self-consistent, only you have to use forms of reasoning that go beyond traditional arithmetic. That was one of Gentzen monumental achievements.
But keep in mind all of this is within the (relatively) small confine of mathematical logic. Some mathematicians have doubted that this was the best approach for mathematics (most famously Poincaré and after him Brouwer and the intuitionists). Although it is one kind of the default approach (even though in reality mathematicians don't bother too much with fondations) there has been, paradoxically, a formalization of intuitionist logic, which is a generalization of classical formal logic, and it turns out to be pretty important in the very large topoi theory.

>>13552083
Interesting stuff. What are your thoughts on some of the attempts of reconciliation like Leibniz's Characteristica Universalis or Langan's CTMU >>13551687. Do you think there's headway here and if so, will the community be receptive?

>Fwiw you can prove traditional arithmetic is self-consistent, only you have to use forms of reasoning that go beyond traditional arithmetic.
Were these transformable? Or was it incompatible with traditional mathematic logic only could only function in their own spheres of logic.

btw, you;re the MVP of the thread anon. Lot of interesting ideas to delve into.

Absolutely A+ thread. As for actual book recommendations, Frege's Foundations of Arithmetic is an absolute classic, and it's very readable even to non-mathematicians. For others, I find Boolos to be an excellent read, though I have strong logicist tendencies so that might color it. Logic, Logic and Logic is one of his best and most accessible books. Gödel's philosophical papers are also a treat, even beyond his staggering mathematical achievements.

>>13552183
>on some of the attempts of reconciliation like Leibniz's Characteristica Universalis or Langan's CTMU
I haven't dwevled in them too much, I only read a few summaries of the CMTU written by Langan himself. Those attempts are interesting to the extent they're carried out seriously, but I doubt they can settle definitely any real philosophical problem.

My takeaway from Gödel's incompleteness and the subsequent developments (including Gentzen's result among others) is precisely that logical constructions rest, in last analysis, on philosophical choices. Gentzen uses recursion up to espilon-0 to prove recusion up to omega-0 (traditional arithmetic recursion) is consistent. What does that mean? Should one think that means arithmetic is consistent enough, or does it show a lack in arithmetic that one has to resort to stronger recursion? What about the fact the Gödel's result only talks about Peano arithmetic (traditional formalization of arithmetic)? Why should we care particularly about that one arithmetic (because it includes all we mean traditionally by arithmetic, but why is that traditional meaning so important?).

Ultimately it boils down to what you consider appropriate or interesting. From this consequences follows, but the initial impulse is rather unjustifiable, or rather we hardly bother justifying it (not to mention it seems impossible).
So I have little doubt that Langan's construction is formally stasfying. But what does it bring besides the aesthetic pleasure specific to the mathematician? Why insist on having a single principle and no assumption? The love of unicity often leads the intellect astray in science, and if the CMTU is not science, why should we take it seriously as a account of all reality? I'm afraid it will only satisfy someone who's only looking for a pretty system.

The devil here is in the underpinnings. By the time you've written the first definition you already have made the most important decisions, often without telling the reader. Any real philosophical take imo should be precisely about how and why we make those decisions. Not to mention it's often more impulse than decision.

>>13552218
strong smell of onions from this post

>>13551347
Calvin and Hobbes is the single greatest cultural production of the 20th century. Prove me wrong pro tip you cant

>>13551306
The Tao of Physics

>>13552183
Forgot to answer the rest of your post.

>Do you think there's headway here and if so, will the community be receptive?
I doubt the community of logicians will care much for it, it is suspecting of outsiders and Langan's description strike me as a bit self-indulgent. Logicians rarely attempt such ambitious undertakings, precisely because history has made them painfully aware of the limitations of logicism.

I'm partial to phenomenology and even sociology because it allows to discuss degree of arbitrariness in those systems.
But I should give Langan credit for having tried built a perfectly aristotelian system. In an other age (Leibniz's age) it could have been huge (but again category theory and the like weren't fully formed at the time).

So to sum up, I take issue with his claim that having a sound logical system is enough to encompass reality, which is the central issue here (logic/reality difference), but I can admire the architectonic character of the result.

>Were these transformable? Or was it incompatible with traditional mathematic logic only could only function in their own spheres of logic.

I'm not sure I understand the question very well, but if you ask whether those means could be translated into usual arithmetic, then no, it went infinitely beyond it, but contained it in a way. To be more specific: one of the key element of traditional arithmetic (and this is reflected in Peano's axiomatization) is the principle of recurrence, already used in Euclide: if I have proved a proposition holds up to a integer number n, and I have also proved that if it holds for n, then it holds for n+1 the number immediately after it, then those two facts together give me that the proposition holds for any integer number, no matter how big.

Sounds pretty natural, but the problem is there are an infinity of integer numbers, so you need a principle to guarantee that the transition can hold ad infinitum.
In traditional arithmetic it is considered legit, hence the recurrence principle. But some people think we should only reason on finite numbers we can manipulate physically (by counting on your fingers or in a computer for instance).
What Gentzen use is a much more powerful recursion principle. Say for instance you have an infinity of propositions to prove, and each proposition depends on the proposition before, and each applies to all integers. You can use recursion in the usual sense for each of them, provided all propositions before have been proved. But can you apply recursion to the infinite succession of infinite proposition? It's not clear this is allowed by the usual recursion principle. An analysis might show it boils down to the usual, but there are more complicated way of imbricating recursion that are definitely not allowed by usual recursion. Gentzen used such an imbricated recursion.


>btw, you;re the MVP of the thread anon.
I think that honor goes to >>13551389 and his subsequent posts but I'm still genuinely flattered.

>>13551806
>I would recommend that you just start learning math, even at it's most basic level.
That's the hope, but I struggle with it immensely and I hope that approaching it philosophically may serve as a kind of quiet backdoor that will make it more identifiable and thinkaboutable to me

>>13552319
I wouldn't want to try. It's only fault is that /reddit/ can glom onto the lesser strips and pretend that it was written for ((them)).

That's not really Waterson's fault though.

>>13552516

>>13552531

>>13552257
eat shit nigger

>>13551684
>and it derives from the original human culture - the Bushmen of Africa who were the original Pythagorean-Taoists
WE WUZ PYTHAGOREANS N' SHIET

>>13551785
T. brainlet

>>13551824
bro do you have like, a twitter or something

>>13551306
Blüthenstaub
The Fourth Dimension: Sacred Geometry, Alchemy, and Mathematics
The Metaphysical Principles of the Infinitesimal Calculus

>ctrl + f badiou
>0 results
sad

>>13553372
https://www.jstor.org/stable/10.1086/660983
https://www.jstor.org/stable/10.1086/662746
https://www.jstor.org/stable/10.1086/662748

>>13552499
>

>>13551389
If you don't like the foundations, create better ones yourself.

>What is the epistemological status/importance of someone saying "No, I don't think A=A."
Irrelevant.
You can reject any Axiom of ZFC, that you want to reject.

>We're talking from math to reality, rather than the other way around.
And that is a good thing.

>>13551389
heh...nice try math... but what if empty sets don't actually exist checkmate.

>>13551965
Mathematics is about developing tools to effectively describe observed phenomena, you don't "observe it" like that.

Richard Courant's "What is Mathematics?" deserves to be mentioned (and read).

https://plato.stanford.edu/entries/philosophy-mathematics/

>>13551306
Wilbur Knorr - The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry.
Paolo Ruffini - Teoria Generale Delle Equazioni: In Cui Si Dimostra Impossibile la Soluzione Algebraica Delle Equazioni Generali Di Grad Superiore Al Quarto (this book basically kickstarted mathematical formalism in general and abstract algebra in particular).
Kurt Godel - Collected Works.

Isn't this what Kant's whole synthetic a priori is about?

>>13556936
no

>>13551306
Apostol Calc.
alternatively, Gravity's Rainbow.

>>13551306
Search up the Syntopicon get the online version. Search up all mathematical terms.

bump

Corollary to my OP, how do I learn basic math aside from philosophical considerations as a carrot on a stick? I want to learn calculus, but even just algebra is a struggle for me.