Klein: Greek Mathematical Thought

[Anonymous No.21887744]

I don't get it. It feels like a semantically-empty exposition that is shaped in the image of insight.

[Anonymous No.21887749]

Could be yeah gays

[Anonymous No.21887772]

>>21887744
Well, what do you take his basic thesis to be?

[Anonymous No.21887794]

>>21887744
Can you explain, without looking it up, what a division ring is? It's important to assess if this conversation is worth having.

[Anonymous No.21887799]

>>21887772
That the Greeks had a more "natural" ontology of number that we no longer understand thanks to the symbolic revolution of algebra carried out in the Middle Ages and the Renaissance. We no longer think of number as beings but rather as an abstraction of an abstraction (symbols representing abstractions). The thing is, I don't know how number can have its own "being" because it's a predicate of something. You can't point to "seven" in the world without pointing to "seven" *things* (that are not seven themselves). And even if you see number as a relation, relations aren't beings. I'm either confused or this entire work is confused (probably Heidegger's fault).
>>21887794
I couldn't. I've only taken up to multivariable calc, linear algebra, and probability. Dearth of proofs in my education.

[Anonymous No.21887820]

>>21887799
Do you take it as having a prescriptive aim which you find you disagree with?

Let's bracket whether number has being in the way described and just ask: does he show that the Greeks believe this, whether mistaken or not, and that this is different from arithmetic from Vieta on?

[Anonymous No.21887838]

>>21887820
I don't have a prescriptive problem with it. The worldview he's trying to prove genuinely doesn't make sense unless it's an overwrought yet under-realized idea that needs further explication. To prove the difference, the ontology has to be (at least somewhat) coherent first. Could they have believed this?

[Anonymous No.21887851]

>>21887799
>That the Greeks had a more "natural" ontology of number that we no longer understand thanks to the symbolic revolution of algebra
Well, that's a historical ---and philosophical--- fact. Greek ontology of number was organic (integrated in the movements of geometry); algebraic numbers are abstract.

[Anonymous No.21887854]

>>21887851
How is geometry not an abstraction itself?

[Anonymous No.21887883]

>>21887838
So if may state what I take Klein's interest to really be, is less whether the Greek view is true (I say "less" and not "wholly uninterested" since it's still worth evaluating), but rather to bring to light a perplexity about modern science, which sometimes fancies itself as strictly empirical, but which makes most of its claims about nature via this more abstract math from Vieta and Descartes. He's trying to set the ground down by this kind of study for a renewed questioning (and not necessarily wholesale doubt) about what we can say we know by mathematical physics if it has this abstract character in discussing the things of the world.

>>21887854
Nta you're responding to here, but while the proofs have a more abstracted character, isn't it the case that geometry holds remarkably well for understanding the physical space we experience? That is, if you're trying to find the double of a roughly square space of land, doesn't marking out the square that emerges from the diagonal of the starting square still get you the double you practically needed? This is before even getting into how instantiations might fall short of "perfect shape".

[Anonymous No.21887890]

>>21887883
Wait until you start measuring coastlines.

[Anonymous No.21887896]

>>21887799
>That the Greeks had a more "natural" ontology of number that we no longer understand thanks to the symbolic revolution of algebra carried out in the Middle Ages and the Renaissance. We no longer think of number as beings but rather as an abstraction of an abstraction (symbols representing abstractions).
No. You've got some things right and muddled others. But in any case, whether you agree or not with what you wrote as you see being the thesis is irrelevant to the argument of Klein's thesis as they differ in fundamental ways. Go read the book again and when you've gotten in right, come back.

>Oh, well why don't you just tell me what's the correct thesis
No, if you want to further examine any philosophical groundings for mathematics then the least you can do is get the thesis, or even just a layman's summary correct.

>The thing is, I don't know how number can have its own "being" because it's a predicate of something. You can't point to "seven" in the world without pointing to "seven" *things* (that are not seven themselves). And even if you see number as a relation, relations aren't beings. I'm either confused or this entire work is confused (probably Heidegger's fault).

My advice is to put this book down and go take any introductory philosophy of maths course available. That advice may make you upset, you may seethe and try to insult me as you lash out from a wounded ego, but the fact of the matter is, you will be much better for it if you follow my advice.

[Anonymous No.21887900]

I got a prof that has such a hard on for Klein. Lives and dies by Klein's interpretation of Meno. Might have to give this a read

[Anonymous No.21887902]

>>21887896
I just don't have the time or the resources for that. If you want to give me decent advice, give me something that's practically achievable. It just sounds like you're gatekeeping for the sake of vanity.

[Anonymous No.21887909]

>>21887900
Having read his two Plato books and the book in OP, I think the OP book is his best work.

[Anonymous No.21887921]

>>21887896
I'm the anon who asked OP to restate what he thought the thesis was, and I have say that without stating what you think he gets wrong, your advice isn't going to help him, since what's he suosed to do, re-read with the same conclusion without it being clear where he might've gone wrong? What feels like handholding to some people can be irritating, sure, but pointing out a few concrete points of failure to understand, as you see it, gives more direction.

[Anonymous No.21888049]

>>21887883
>>21887890
for reference
>>21887851
>Greek ontology of number was organic (integrated in the movements of geometry)
See, this is what I mean by
>semantically-empty exposition that is shaped in the image of insight.
What the fuck is a "movement of geometry"?

[Anonymous No.21888147]

>>21888049
But measurement per se is separate from geometry, which talks instead of what will be true of already determined measures with respect to others; measurement per se requires someone to say "I measure this length in accordance with the king's foot". Unless you have something more specific in mind, geometry as the setting forth of what's true about the relations between space and shape doesn't see to be affected by the paradox, or at least I'm missing what in Euclid, for example, would be a problem.

>What the fuck is a "movement of geometry"?
Presumably anon has in mind Ptolemy or ancient astronomy as the application of geometry to heavenly bodies in motion?

[Anonymous No.21888165]

>>21887799
allow the great John Gabriel to explain:
https://www.youtube.com/watch?v=LOQRDRLvQz0
https://www.youtube.com/watch?v=YXIxWlePBgU

[Anonymous No.21888299]

>>21887799
i dont think numbers were quite as abstract as you say in the middle ages. they very much dealt with practical cases. in the Al-Jabr the author wrote down 6 equations instead of the usual ax^2 + bx + c = 0 that we write now, and for each case gave an example why and how it is used for
some would say the abstract and disconnected view of numbers is more because of the 1900's and the set theory revolution

[Anonymous No.21888314]

>>21888049
how is this a paradox, this is as stupid as trying to measure the circumference of a circle with straight lines. if you have a complicated geometric shape then the length of your basic measuring tool if you try to count it step by step if ofcourse going to be wrong, you are explicitly approximating

[Anonymous No.21888325]

>>21888314
>>21888147
alright, idk if you're the same poster or not, but we already had somebody try to defend the "naturalness" of geometry here >>21887883 when they were really describing a process of measurement

>how is this a paradox, this is as stupid as trying to measure the circumference of a circle with straight lines.
why is it stupid? aren't circles claimed to be natural objects?

[Anonymous No.21888331]

>>21888147
alright, idk if you're the same poster or not, but we already had somebody try to defend the "naturalness" of geometry here >>21887883 when they were really describing a process of measurement
>>21888314
>how is this a paradox, this is as stupid as trying to measure the circumference of a circle with straight lines.
why is it stupid? aren't circles claimed to be natural objects?

[Anonymous No.21888357]

>>21887883
>So if may state what I take Klein's interest to really be, is less whether the Greek view is true (I say "less" and not "wholly uninterested" since it's still worth evaluating), but rather to bring to light a perplexity about modern science, which sometimes fancies itself as strictly empirical, but which makes most of its claims about nature via this more abstract math from Vieta and Descartes. He's trying to set the ground down by this kind of study for a renewed questioning (and not necessarily wholesale doubt) about what we can say we know by mathematical physics if it has this abstract character in discussing the things of the world.
so... is he recapitulating Kant's Critique of Pure Reason but throwing math in the same league with speculative metaphysics?

[Anonymous No.21888361]

I don't have time to read the thread but I think it's important to understand that Klein's motivation and intellectual background is in the phenomenological project which its desire to break through the accretions of modern philosophy and the modern sciences and go back "to the things themselves!" (Sachen selbst!), meaning to the "primal" phenomena (ta phainomena), i.e. the appearances just AS they appear to us, PRIOR to any interpretation or "sedimentation" (key phenomenological term) of interpretations that have become unconsciously habitual.

Klein's project isn't to assert the superiority of Greek geometry, it's basically to show that the modern tendency, i.e. to subordinate Greek geometry (and thus the Greek phenomenological experience of geometry) to modern geometry by "reading it through" modern geometrical notions and frameworks which are "better," "more complete," etc., is revealing of just how buried (sedimented) modern consciousness is. The problem isn't that the sediments are necessarily wrong or bad, it's that they have become unconscious, habitual.

Once we de-sediment ourselves, and return to the "things themselves," we can then CHOOSE between different forms of geometrical thought, and thought in general. We can USE our frameworks, like the framework of early modern physics which is so bound up in Galilean-Cartesian-Newtonian mechanics and its abstract-geometrizing of nature, and subsequently the framework of abstract algebraic geometry which has removed even the intuitive components and reduced all of nature to the logicistic function-thinking of modern mathematics.

The possibilities of "other" mathematics, of a more primal and phenomenologically self-aware mathematics, and of "other" and more primal and self-aware thinking in general, are thus bound up in the phenomenological deconstruction of the modern mathematical sciences, which, because of their utility and their standards of rigor and their association with ideologies of progress etc., have become the sole and unconsciously accepted criterion of truth IN GENERAL, not just in their originally limited domains (which, recall, were utilitarian applications of mechanics in early modern technological/industrial contexts).

There's a reason Paul Piccone's Marxist reading of Husserl, fusing Husserl's phenomenological de-sedimentation with Marx's Hegelian deconstruction of alienated objectivity, is fairly close to Klein's project. Staying within mathematics, Morris Kline's Jamesian/Whiteheadian pragmatist deconstruction of modern abstract mathematics is very close to it as well.

[Anonymous No.21888429]

>>21888331
I'm the same poster you cite; with the example I gave in the other post about doubling a plot based on a diagonal, I don't take it that the example requires a specific measuring unit for the relations in space to hold true, which was allI was going for. I think geometry *can* be abstract when we're generalizing about given shapes, but it's concrete when applied to whatever space is before you. Does that seem fair to say, or would you still argue that it's abstract in application? (Maybe we would need to clear up what we each mean by "abstract"...)

>>21888357
I don't think so. I think (as per >>21888361, which is a very good summary of Klein's efforts and intentions) that he's trying to ask whether mathematical physics is justified in discovering what's true about nature without merely accepting it dogmatically. I take him as open to it being true, but needing to be certain about what we're actually doing when we proceed by symbolism that replaces a (comparatively) more concrete view of number.

[Anonymous No.21888447]

>>21887744
>It feels like a semantically-empty exposition that is shaped in the image of insight.
>(((Jacob Klein)))
These two things usually go together

[Anonymous No.21888455]

>>21888447
You don't need Jews to do that, Russell was more than capable of just that.

[Anonymous No.21888476]

>>21888455
Russell was British, meaning he was a spiritual Jew (or at least a fellow traveler)

[Anonymous No.21888489]

>>21888429
>I'm the same poster you cite; with the example I gave in the other post about doubling a plot based on a diagonal, I don't take it that the example requires a specific measuring unit for the relations in space to hold true, which was allI was going for.
where is the "number" then, if geometry is all about relations? what happens if you try to measure the diagonal of, let's say, a unit plot?

[Anonymous No.21888490]

>>21888476
>Russell was British, meaning he was a spiritual Jew (or at least a fellow traveler)
>I will not cease from Mental Fight,
>Nor shall my Sword sleep in my hand:
>Till we have built Jerusalem,
>In Englands green & pleasant Land.

[Anonymous No.21888569]

>>21888489
>where is the "number" then, if geometry is all about relations? what happens if you try to measure the diagonal of, let's say, a unit plot?
I don't know if we're speaking past each other, but I wasn't talking about whether ancient arithmetic was concrete, but addressing whether geometry is abstract. Do you mean by "number" here something like length measured or qantifiable area of a space? Ordinarily, if you apply a number, it's going to be a measure customarily agreed on, right? But this need not always be the case, for example, in speaking of the length demarcated by a tree and a fence post; take a string of that length, make a square of space roughly, and you can still meaningfully double that area by the diagonal without having to subscribe to a particular measuring unit; for intents, the distance between the tree nd fence pole make up the units of the square.

[Anonymous No.21888571]

First couple of chapters of Spengler's Decline of the West also deal with how mathematic of today differs from that of Ancient Grreks. It might be more your speed, if you find Klein too difficult.

[Anonymous No.21888617]

>>21888569
look man, idk if this is again a different person, but how am I supposed to square what you're saying now with
>>21887851
>Well, that's a historical ---and philosophical--- fact. Greek ontology of number was organic (integrated in the movements of geometry); algebraic numbers are abstract.
Is geometry concrete or is it abstract? Is number concrete or is it abstract? Is geometry connected to number or not? If I could pin whatever you're discussing to a set system of claims, then I could have a fruitful conversation. That's the problem I'm having here. I'm *deeply* interested in what Klein has to say after delving into Heidegger, late Plato, etc., but I have no idea whether he's actually positing anything with any semantic content because he's playing around with so many ideas at once without committing to any of them. Same here goes with you. There's a reason why I'm asking you to consider the diagonal of a root square WRT the question of geometry's connection to number. whether geometry is "organic", whether measurement can be considered a "practical" geometry, etc. Because there are plenty of similarities here, I get that, but then you have problems such as incommensurability that demolishes any ontological coherence between the two.

[Anonymous No.21888636]

>>21888571
>First couple of chapters of Spengler's Decline of the West also deal with how mathematic of today differs from that of Ancient Grreks
Spengler is a retard who knows fuck all about mathematics.

[Anonymous No.21888723]

This book and many others are beyond my reach forever...I struggle with following logical arguments, I dont understand the proof of the irrationality of root 2, the Pythagorean theorem, the rule of three or adding fractions. In the end it doesn't matter as I teach meditation for a living.

[Anonymous No.21888886]

>>21888617
I'm a different anon than >>21887851, but if it's not too presumptuous to speak for both myself and him, we're both taking geometry to be in some respect concrete, and Klein's book seems to be arguing that Greek arithmetic was also concrete (even if, if you'd prefer, in a comparitive way), and that the "numbers" of Greek arithetic are different than the treatment of number in symbolized algebra. I've been focusing more on the issue of whether geometry is abstract, hence I've been avoiding number as an issue that distracts from that. Busy atm, but I'll hop back in later.

[Anonymous No.21889665]

>>21888886
I'm willing to entertain the idea that geometry is, in some aspects, "concrete." But it is still at least a layer or two of abstraction from the reality we experience.

[Anonymous No.21889726]

Anyone who likes Klein should really know about Wildberger, just in case they don't already. Go into his playlists and find any of his full courses, particularly the one where he begins by giving full constructive proofs for the fucking background of Euclid's Elements by starting with straight-edge constructions from before they even introduced the fucking compass yet. He shows how much you can proveably construct JUST with a straight edge. Wildberger is incredible.

[Anonymous No.21889967]

>>21889726
Klein, Guenon, Heidegger, Wildberger, etc., all form an interesting anomalous little bubble in Western phil together

[Anonymous No.21890872]

bump for maths

[Anonymous No.21890931]

>>21889967
Wasn't Guenon a math dropout? Why should anyone take him seriously?

[Anonymous No.21890936]

>>21890931
because he had interesting things to say about infinity that make sense given the hundreds of years debate on the philosophical foundations of calculus (that were never really resolved desu)

[Anonymous No.21891090]

>>21887744
>semantically-empty exposition that is shaped in the image of insight
Like all prominent modern "philosophy"

[Anonymous No.21891251]

>>21888723
If you can learn how to work this site's UI and get past captcha you can learn to add fractions.

[Anonymous No.21891259]

>>21889967
>Guenon, Heidegger
I was just thinking of the relationship between these two this morning. I think Heidegger was a Traditionalist but he didn't even know it. He all but states it here: https://youtu.be/4WK8PJvkzG0

[Anonymous No.21891293]

>>21891259
Heidegger is more "trad" than Guenon. Expect a thread on him in a minute. I think /lit/ is ready for this discussion.

[Anonymous No.21891309]

>>21891293
Would be a real synchronicity if that happens because I was literally thinking about making such a thread with my morning coffee today. Decided otherwise as I'm not well versed enough in Heidegger, but feel free to do so if you are.

[Anonymous No.21891319]

>>21891309
Synchronicity, or evolution of misfit intellectuals looking for answers? Who knows. All I know is I'm glad I have frens who care about the same things I care about.

It is done. >>21891310

[Anonymous No.21891716]

>>21888723
>I dont understand the proof of the irrationality of root 2
Do you know prime numbers and can you do primfactorization of numbers? Do you get what a proof by contradiction tries to achieve? If you get these two covered you can understand the proof. If you want to understand math have fun with math. Do for example factorizations for shit and giggles. Or when you see a number try to express it in algebra like 32 is 2^5

[Anonymous No.21892506]

brine bumperization

[Anonymous No.21892973]

cumpanino

[Anonymous No.21893906]

bumpman

[Anonymous No.21894648]

he is up to something

[Anonymous No.21895700]

Klein was retroactively refuted by Whitehead

[Anonymous No.21896174]

bump

[Anonymous No.21897439]

filtered