On 10 Mar 2014, at 16:16, Gabriel Bodeen wrote:



On Saturday, March 8, 2014 2:37:50 AM UTC-6, Bruno Marchal wrote:
A couple other accounts of how things might be that I take seriously are (1) physicalism in the sense that arithmetical propositions might only be true when physically realized,
No problem, and indeed this would make comp false. of course, if you really defend that thesis, you have to explain and prove the existence of infinitely many prime numbers by using physics, and this without presupposing addition and multiplication of integers. I am not even sure how you will just defined what is prime number.

Given a correspondence theory of truth and this kind of physicalism, mathematical theorems would be true if and only if there's a corresponding physical reality.

That is too much vague for me. I can interpret this in too much sense.




So, for example, if the universe is finite, then there wouldn't be infinitely many prime numbers.

This is non sense. In my humble sincere feeling.

Even if physicists get a knock down argument in favor of a finite universe, that would not refute at all Euclid's theorem that there is an infinity of prime.

A prime number is just not a physical object.





Nor would there be infinitely many integers.

That is ultrafinitism. That is why I make "arithmetical realism" sometime explicit.

In that case indeed we are out of the scope of my expertize.
But if step 8 is correct, that moves will still prevent you to say "yes" to the doctor, unless more and more ad ptolemaic redefinition of "matter".





But there would still be integers and primes.

Thanks for reassuring me. I was about to close Platonia for bankruptcy :)




Numbers, addition, and multiplication would be patterns that our brains recognize in material things,

With Church thesis, we have a notion of universal machine which generalizes this, non trivially. Comp makes obvious the use of those mathematical tools.





at first due to experience counting objects, grouping them, and counting groups. We abstract those patterns to symbolic form in our heads or our writing for convenience, and we generalize the notation to cover a wide variety of patterns. But our process of abstracting and generalizing may omit important limitations (such as finitude) of the physical reality on which it was originally based.

Assuming a physical reality at the start. For the mind body problem it is better to be, at least methodologically agnostic, about that.

You describe well how humans got the numbers, but it is a projection to believe that the notion of humans is more conceptually simple than the notion of numbers.

The question is only, do you agree on the axioms I gave.

With comp, your "numbers are a human cultural construction" becomes "numbers are universal machine cultural construction.




or even (2) relativism in the sense that arithmetical propositions might only be true for humanlike brains,
OK, but same remark. Defined human-like brain, and give me a proof that 1+1=2 from that definition.

Due to our shared evolutionary history, humans share nearly all their brain architecture in common. Due to our shared cultural history, many of the humans we regularly encounter share much of their set of background assumptions and beliefs in common. It appears that there's no such thing as such a perfectly lucid and detailed description or set of instructions that one person could give another that eliminates the need for the other person to grok the meaning, i.e. to connect the ideas appropriately and fill in the missing information based on their own "wiring" and their own experience. (Take Edgar as an demonstration of this apparent fact. ;)


That's why you publish, or put down thesis. You have to play the academic game. The academic is the worst of all systems, except for all the others (to parody Churchill).






Consequently, some suppose that communication with an extraterrestrial intelligence may fail due to there being an almost total mismatch in "wiring" and experience. (When language fails, we humans resort to pointing at objects and pantomiming, but without shared sensory systems and emotional responses, even that may well fail to be grokked by the alien.) It's also possible that the symbolic structure of our mathematics is dependent on our "wiring" and experience; indeed there is some evidence that the way humans use language is due to an evolutionarily recent genetic mutation.

That's a computationalist type of explanation, no problem.





For these two reasons, our mathematical definitions, theorems, proofs, etc may only be suitable for use by other humans, and so by a pragmaticist alethiology, only true for humans.

?
If comp is true for human, that's what counts. It means that they can survive through a relative universal numbers, but then, without adding magic, what is true for all universal numbers is true for human true, and universal numbers have an amazing theory of self-reference.




The usual proofs then apply, because we're humans.


Cool.



with an alethiology of the sort preferred by the American pragmatist school of philosophy.
keep in mind that you mention people who are Aristotelian, and the point I do is only that IF comp is true, THEN such approach get inconsistent or epistemologically non sensical.

Hm. Can you elucidate what you mean by saying they are Aristotelian? What is the key contrast?

The Aristotelian believes/assumes a primitive material universe.

The Platonists believes/assume the idea that what we observe is the shadow of something else.





Four options plus an ignorance prior and little evidence gives me about 25% confidence for each. :)
ONLY IF you develop your alternate assumptions. The idea that "1+1 is prime" independently of human is far more simple (and used) than the idea that "1+1 is prime" is relative to the human brain. The axiomatic of natural numbers is far more simple than anything else. You can always propose a much more complex theory to falsify a simple set of axioms.

I don't know that the other cases I've mentioned are more complex. Physicalism just puts some mysterious "matter" first and makes math derivative of it. That may be wrong, but it's hard to see why it's more complex than comp's reversal of it.

Comp assumes only the numbers, and the law of addition and multiplication. Of the combinators with the laws of the application and reduction.

We search for an explanation for matter and consciousness.




The relativism described above isn't an additional supposition added to math; it takes ideas from biology and linguistics to see what consequences there might be when they intersect with math.

That relativism argues against comp, and even implicitly against Church thesis. But my point is not that comp is true, just that with comp, the theory QM + comp is redundant, and we have to justify QM (at the least its logic) from self-reference. And up to now, it looks it works.

Bruno




-Gabe

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