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B Would Mathematics be Considered a discovery or an Invention?

  1. Sep 15, 2017 #41

    jbriggs444

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    The point I was trying to make is that whether the value you choose to label as 1 in your system of natural numbers happens to be ##\frac{\sqrt{2}}{2}## baryons, ##\pi## apple pies, 12 flying purple spaghetti monsters or one set containing an empty set does not change the nature of the resulting system of natural numbers. The choice has no physical consequences. It does not change the natural numbers. Nor does it change quantum mechanics.
     
  2. Oct 15, 2017 #42
    I wanted to comment on it a little bit, but I was unsure how to do it in a more coherent manner (without diverging too much). First a little off-topic comment about Kronecker's quote. I remember reading a more extensive version of it where he basically says something along the following lines. What he was saying that if he had enough time (which he didn't) from his main mathematics, he would independently develop an entirely different set of ideas from scratch.
    I tried to find that extended version but somehow I just can't find it now. I have learnt that the original quote is also supposed to be recorded/taken from some lecture. So I am not sure whether the extended version which I read was correct or not.

    Secondly, as I understand, there are basically the following ways to see the problem (as we go down, roughly speaking, the philosophical importance decreases and practical importance increases):
    (1) Saturation Problem (Generalised Church Thesis)
    (2) Church Thesis (Bounded Memory)
    (3) Efficiency of Algorithmic Calculations


    The question (2) is settled in an absolute decisive sense (and hence also the absolute nature of incompleteness in the "bounded memory" setting). This is as long as one doesn't conflate (2) (a mathematical statement) with Physical Church thesis (a physical statement). There have been few proofs of (2) based upon axiomatisations (I don't know much in the way of details). Though perhaps not everyone might agree that a given axiomatisation is absolutely convincing.
    But actually, I still believe that the truth of (2) is absolutely decisive (at any rate). I say this because of the growth functions that are formed as a result of the underlying hierarchies (by extending unsolvability). And also based upon how our natural understanding of "limits" of (transfinite) iteration (ordinals that is) happens to coincide with its recursive counterpart (that is using the symbols of ℕ only). So the words "bounded" and "recursive" happens to coincide exactly in this context. This is a point I happened to stumble upon a few years ago by "trial and error" before I first saw the term "ordinal" being used. But I don't know what would be the best way to make this particular point more formal.

    Though I am not trying to downplay the importance of precise axiomatisations at all (in case it seemed like this), the two points in previous paragraph leave no doubt whatsoever in my mind that (2) is correct.

    ==============================

    https://www3.nd.edu/~cfranks/frankstennenbaum.pdf
    Specifically I wanted to quote the following part (on page-9):
    "We might be inclined to doubt the finitist character of the ‘transfinite’ induction [through ε0 used in his proof of the consistency of Peano Arithmetic (PA)], even if only because of its suspect name. In its defense it should here merely be pointed out that most somehow constructively oriented authors place special emphasis on building up constructively . . . an initial segment of the transfinite number sequence . . . . And in the consistency proof, and in possible future extensions of it [to theories stronger than PA], we are dealing only with an initial part, a “segment” of the second number class . . . . I fail to see . . . at what “point” that which is constructively indisputable is supposed to end, and where a further extension of transfinite induction is therefore thought to become disputable. I think, rather, that the reliability of the transfinite numbers required for the consistency proof compares with that of the first initial segments, say up to ω2, in the same way as the reliability of a numerical calculation extending over a hundred pages with
    that of a calculation of a few fines: it is merely a considerably vaster undertaking to convince oneself of this certainty . . . ."

    Not counting some more precise technical details involved (which I obviously don't understand), one can still get a fair sense of the paragraph.
     
    Last edited: Oct 15, 2017
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