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September 6th, 2018, 09:42 AM   #11
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Yes! Exactly my criticism: Their argument is viciously circular. I am currently writing a paper about brains and learning where I criticize these authors, precisely on that ground, but I also wanted to refute their alleged "elementary mathematical truth" (whatever the hell that means), and currying seems to me to provide a compelling, if not definitive, refutation. I just can't believe this book got published by such a prestigious publisher. And Gallistel, well, he's a big name in the psychology of learning.

Thank you!

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Jose
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September 6th, 2018, 01:57 PM   #12
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Yes! Exactly my criticism: Their argument is viciously circular. I am currently writing a paper about brains and learning where I criticize these authors, precisely on that ground, but I also wanted to refute their alleged "elementary mathematical truth" (whatever the hell that means), and currying seems to me to provide a compelling, if not definitive, refutation. I just can't believe this book got published by such a prestigious publisher. And Gallistel, well, he's a big name in the psychology of learning.
Oh well then you and I are on exactly the same page. I didn't mean to go from explaining the math to editorializing about their thesis, but now I'm glad I did. What they're doing is starting from the assumption that the brain is a computer in the sense of a Turing machine; and from that, work out their ideas.

Still, currying doesn't violate what they say, because one of the functions in currying outputs other functions, and they are saying you can't break down a function to functions that output variables. So you need to be careful there.

But their thinking is way too simplistic. They're considering functions of a finite number of variables. But physics uses infinite-dimensional spaces to do quantum mechanics, and QM surely underlies biology and whatever "computations" are being done in the brain. So they are not really taking the full range of science into account. They're trying to fit the brain into a very simplistic model of math, functions that input points and output points. In QM, the "points" are actually complicated functions themselves. We take inner products that are analogous to the dot products of basic vector calculus, but in QM the inner product is an integral of the product of two functions. Their model doesn't consider this. Their thinking is just way too simplistic to be meaningful. I'm sure they're very smart people. I share your feelings about the amount of nonsense that gets written by very smart people.
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September 7th, 2018, 06:03 AM   #13
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Thanks for that clarification. So, are they correct in stating that "functions of two arguments, which include all of the basic arithmetic functions, cannot be decomposed into functions of one argument" (p. 53)? Again, they call this an "elementary mathematical truth" and a "logical, mathematical fact about functions" which sounds suspiciously too strong to me. They give no demonstration of this (they only give two examples, but this hardly qualifies as a demonstration). Has anyone demonstrated this? If they are not correct, and if currying does not refute them, what does?

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Jose
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September 7th, 2018, 09:46 AM   #14
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Thanks for that clarification. So, are they correct in stating that "functions of two arguments, which include all of the basic arithmetic functions, cannot be decomposed into functions of one argument" (p. 53)? Again, they call this an "elementary mathematical truth" and a "logical, mathematical fact about functions" which sounds suspiciously too strong to me. They give no demonstration of this (they only give two examples, but this hardly qualifies as a demonstration). Has anyone demonstrated this? If they are not correct, and if currying does not refute them, what does?
I don't know. I'd focus less on the specifics and more on the irrelevancy of the observation to anything important. The brain doesn't implement functions like a computer does. And most functions aren't computable. I'd approach it from that angle. Otherwise you get bogged down along with the author in their own irrelevant point. Also I wouldn't call that an "elementary mathematical truth." That sounds suspiciously like an evidence-free value judgment. Is the axiom of choice an elementary mathematical truth? It's a basic axiom of mathematics, but it's not elementary and nobody knows if it's true.

Last edited by Maschke; September 7th, 2018 at 09:48 AM.
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September 10th, 2018, 02:36 PM   #15
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What do you mean by "decompose?"

How can one talk about something without knowing what is being talked about?

Do you mean for example f(x,y)=h(x)+g(x)?

(Calculus?)
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September 10th, 2018, 02:54 PM   #16
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How can one talk about something without knowing what is being talked about?
Ahem.
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September 10th, 2018, 04:12 PM   #17
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Ahem.
Applesauce

But strange sequence of events. Always work in silence, but turned on TV since I didn't think this was going anywhere in my favor and accidentally came across a Rachmaninoff piano concerto. Then looked up function decomposition (an engineering term which has nothing to do with OP) and since I was slumming I googled Rachmaninoff Piano Concerto #2- on youtube for cryin out loud.

Excuse me. Ahhhh.

Last edited by zylo; September 10th, 2018 at 04:15 PM. Reason: 2, not 3
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