what is "model theory"?

Model theory (thought not described as such in the book) is a method of description used in the "freshperson" maths book from 1960, Birkhoff and Maclane's "A Survey of Modern Algebra". For example, a group is described as a set of elements together with a set of operations on them.
Birkhoff and Maclane define the concept of "relation". A function is a particular case of a relation.

It is not surprising that this concept of relation was adopted by some computer writers, since many of these people would have studied treatments of maths similar to Birkhoff and Maclane.

Without the work of Tarski, it is no doubt correct that Birkhoff and Maclane's book would have not been written in the form it was. If the word "relation" has not beeN used in maths books in 1960, it seems very likely that it would not have been used in the 1970's in computer books, with the same meaning.

Hence the philosophy professor at Stanford pointed out that the Oracle buildings would not have existed, without the work of Tarski.

One might also point out that the buildings would not have existed without contributions to beam theory by Galileo, Bernoulli, Euler.

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richard mullins

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• Euler
• Birkhoff
• Maclane
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[richardmullins44]

okzoia. I agree with your comments. There are an almost infinite number of different objects we could define.In this way, we could define an almost infinite number of university maths courses. In the same way, we could write an almost infinite number of books, or an almost infinite number of grammars. But how many such objects is it healthy to define and think about?

[richardmullins44]

Thanks okzoia.

[nortexoid]

so how is this video about model theory? i mean, you talk a little about tarski, the founding father of "model theory" (though his form of model theory is different from most modern presentations), and then you talk a bunch of shit for awhile, and then what else? you get off on pretending to explain something complicated, i take it?

[richardmullins44]

@nortexoid You could be right. I saw books on "Model Theory" and could not understand them at all. Then a few years later I noticed that the term "model theory" is used to describe the approach exemplified by Birkhoff and Maclane in their freshperson book "A Survey of Modern Algebra". I found B&M easy to understand, so I suspect that the more complex maths books are failing to present material in a way that is intelligible to the beginner. In 1950 Turing predicted this...

[richardmullins44]

@nortexoid In 1950 Turing predicted this - he was only talking about people who write computer code. But I since begin to think that this disease of obfuscation exists also in many other areas. Perhaps Maths and Philosophy is where it started. It has now spread to many areas such as "business management".

[xmarlasingerx]

I'm asking what the Banach--Tarski paradox is - and why can't I wrap my head around it?

[richardmullins44]

Thanks for your message. (I liked watching your videos too). I haven't studied maths for 40 years. Without funding, I lost interest, or perhaps did not have time.

I think the Banach-Tarski paradox says you could cut up a ball and make from the pieces, two balls of the same size.

This suggests that there is something wrong with the theory. It does not sound like a very useful theory for describing objects in our world.

Are you studying math?

[Maxmuetze]

Banach-Tarski: Given the axiom of choice (!), there exists (!) a partition of the unit ball into five disjoint subsets such that the whole unit ball may be recovered by taking the union of only 3 members of that partition after rotating each in a certain way - as well as by joining just the (appropriately rotated) other two. This merely tells us that our notion of volume does not consistently extend to ALL subsets of space (and in fact those subsets defying it are not explicitly constructable).

[okzoia]

Mathematical "objects" don't have any properties. There are no such things: it's all in the mind - which means we imagine them along with round squares, Pegasus, and hypercubes. "They" are what we make them.

[jstarret]

Banach-Tarski is not about objects in the real world. It is about mathematical objects, which sometimes have paradoxical properties. Sometimes you can modify the properties of a mathematical object to remove a paradox, and sometimes you can't. If you want to keep the axiom of choice, you're stuck with Banach-Tarski.

[xmarlasingerx]

Fun stuff. As for the theory being "useful" - I couldn't agree more. I'm all about useful beliefs and theories - and the application thereof. I think I'll be okay letting go of this paradox; that is, not needing to understand it. :)

[xmarlasingerx]

Argh - I hate how youtube totally just ate my comment. So no, I'm not a math major - I'm the daughter of a Chemist/Math Enthusiast (check out his videos - AncalagonxThexBlack) and actually a Journalism Major myself. I'm just a masochist in that I enjoy reading about and researching mind blowing paradoxes in the company of my roommates.