[Thrash]

Ok.

What is this?

I know I could look up something on wikipedia, but I would rather have a discussion....

Sent from my MB520 using CGR Forum

[Chrysostom]

In the 19th century, mathematicians become worried about the security of math. For instance, Riemannian geometry challenges Euclidean geometry as necessarily true. So a strategy of reductionism arises: map "higher" maths onto "lower" maths, thus displacing the problem to a more secure foundation. (Hilbert) Argue that geometry is consistent if not necessarily true so long as algebra is, and algebra is if arithmetic is.

Ultimately this strategy brings us to Frege, who seeks to provide the foundations of arithmetic with his logicist project. In order to do this he manufactures modern logic almost whole cloth. Russell would also be inspired by this project, and while he would see Frege failed, in an attempt to provide the foundation for mathematics he would formalize math in a wholly unprecedented way with the Principia. He also failed, Wittgenstein crippling him and Godel nailing the coffin shut.

And so the striking thesis: post-foundationalism was announced not by wooly postmoderns but by hard-nosed mathematicians. The specter loomed for a century and at each step eluded defeat; in the end it could not be stopped.

Some concluding thoughts regarding the subsequent history of Western philosophy. The analytic/Continental split occurs at precisely this time, and I would like to pursue how this was caused by the problems in mathematics. It is certainly telling that Heidegger, the font of the split, would split from Husserl precisely on the issue of quantification, numbers, math. Time is not t; a tool weighs not "3.2 pounds" but "too much for the job." So there are two ways to read Heidegger's move here; after all, could he not rely on math itself for post-foundationalism? One, he saw the writing on the wall for the whole sweep of reducing all knowledge to math (keep in mind the snowballing quantification of all disciplines in the 19th century) and sought an alternative. Two, while his work disagrees with the mathematical specifics it is simply part of the conversation in furthering the agenda manifest in foundationalism; after all, did not Derrida read the gift in terms of a mathematically sealed system of economic reciprocity, i.e. did Heidegger in fact spawn an alternative imagination or did he merely provide a sounding board for the root mathematical tradition?

Analytic philosophy clearly had mathematics at its root. Obviously the foundational succession Frege-Russell-Wittgenstein goes explicitly along the lines of mathematics, but in addition to Frege are analytic grandfathers Husserl and Peirce. Further study would pursue the analytic tradition following Godel as a continuation of the foundationalist agenda. I have not re-checked the source material but I suspect logical positivism could be read as a simple reactionary movement clinging to a mathematical foundationalism. Interestingly, even while Quine decimates Carnap etc. he declares truth "the end of science," still holding onto the hope that it's math all the way down.

Another way of saying it is to suggest that the "linguistic turn" was caused by the failure of a fully foundationalist conceptualization of knowledge. This would be sought as a way of maintaining all the upside of foundationalism in the absence of the Archimedean foundations -- e.g., Tarski's semantic theory of truth keeps most every positive out of the correspondence theory of truth, simply recasting it in logico-linguistic terms.

Finally, the build-up to the 19th-century crisis must be explored. Copernicus initiates abstract geometry as the fundamental idiom of knowledge, and this follows through three centuries. Descartes provides the philosophical mythology (and I don't mean that in a negative way) to bring Copernicus's spark to the rest of knowledge. Spinoza makes this explicit, seeking to found all knowledge under the model of geometrical proof. But there are always problems with actual Copernicanism, as its "perfect" abstract theories don't match the world. Newton is inspired by Copernicus but leans heavily on experimental method, and is thus able to complete the Copernican project.

Over the next century the successes of the New Science would seek to bring this epistemic idiom to all areas of human knowledge. But Newton's reliance on abstract, speculative math in his experimental work is precisely the problem: he depends crucially on infinities but deals with them in incredibly imprecise ways. This produces Cantorian set theory and ultimately the announcement of post-foundationalism in mathematics, therefore challenging the security of the whole of knowledge that had been recast following the Copernican-Newtonian lead.

[Thrash]

Hmmm... interesting.

What I am a little confused about is that I thought Cantorian set theory was seen as a base (dare I go as far as foundation?) of mathematics, but you read it as being ultimately post -foundational. Though, it did challenge the older models dealing with infinities, so the older method of searching for this foundation is undermined. Am I understanding you correctly?

I am glad you included this bit of history. I found it very helpful. I have tried to steer clear of more analytical philosophy for a long time, even though I do like reading Wittgenstein's first philosophical writings.

Sent from my MB520 using CGR Forum

[Chrysostom]

It's one of the crucial pieces of 19th century mathematics, foundational in that respect. And more than correcting other views of infinity that were floating around, it replaces no rigorous thinking about infinity with actual math. What's post-foundationalist about it is that once the Cantorian die is cast I think you can no longer say that it's proof all the way down (i.e. foundationalism); Hilbert and Frege and Russell and on and on tried but they just kept failing, and then Godel put the nail in the coffin.

If you want to know in much more detail it would be tough for me, because I'd start having to talk about specifics of set theory, which gets complicated pretty fast.

[Thrash]

The reason I am interested in this is because I know it is essential in Badiou's ontology. I printed off some articles about set theory and will probably do some more searching so I can understand what exactly is going on. Until then I just feel like I can't evaluate correctly Badiou...

Sent from my MB520 using CGR Forum

[Chrysostom]

Yeah sorry, probably can't help you other than to point you to some math books...

[ICallHimTurtle]

This is definitely my favorite thread EVER on CGR.

[Thrash]

Originally Posted by ICallHimTurtle

This is definitely my favorite thread EVER on CGR.

Haha, wow.

Sent from my MB520 using CGR Forum

[Thrash]

Originally Posted by Chrysostom

Yeah sorry, probably can't help you other than to point you to some math books...

I am setting up a meeting with a math professor at Etbu. But... I will be back!
This thread will never die!

No, but seriously. I want to discuss this more later. I regret not having a more comprehensive knowledge of math.

Sent from my MB520 using CGR Forum

[ICallHimTurtle]

Yes, I might do research just to continually bump on-topic forever.