> On 30 Aug 2019, at 20:53, Philip Thrift <[email protected]> wrote: > > > > On Friday, August 30, 2019 at 10:10:22 AM UTC-5, Bruno Marchal wrote: > >> On 30 Aug 2019, at 04:33, Alan Grayson <[email protected] <javascript:>> >> wrote: >> >> If there are infinities in mathematics, but not in physics or in nature, is >> that a problem? AG > > Is that an interesting problem? I guess so. > > Some theories in mathematics assume an axiom of infinity, like in set theory, > analysis, etc. > > That has often led to paradoxes, but they have been “solved” by diverse > means. So most such theories are considered not being problematic. We can > also show that, even restricted on the arithmetical truth (which has no axiom > of infinite, as all natural numbers are conceived to be finite), adding an > axiom of infinity lead to stronger provability abilities. The set theory ZF > proves much more than the arithmetic theories PA, even on just the numbers > relations. Yet ZF, and actually all effective theories are limited on a small > spectrum of the arithmetical reality. The omega-initial segment of ZF mirrors > PA faithfully. > > In physics, the universe itself could be infinite, without having any > infinite things in it, like the model of Arithmetic (all numbers are finite, > and the set of all numbers is just a meta-concept, not representable directly > in the theory, but still manageable (you can prove in PA that there is an > infinite of prime numbers, by proving > > For x (prime(x) -> It exist y (y bigger than x) and prime(y)). > > Are there actual infinite object in the universe? > > I can prove that if mechanism is false, then there as such object. With > Mechanism, the mind is infinite, and physics is somehow the Mind seen from > itself internally. That might favours an infinite physical universe. Does our > substitution level depend on Planck Constant? Open problem. > > With mechanism, the axiom of the infinite is inconsistent on the ontological > level, but is a theorem on the phenomenological level. It shortens the > proofs, and provide many tools to to handle mathematically the semantic, the > notion of limit, many form of approximation, even to learn just about the > natural numbers or the combinators. > > Mechanism provides a testable account of the mind-body relation, an account > which does not assume more than elementary arithmetic, and which doesn’t > involve any other ontological commitment. So let us see. The quantum > structure, and time, admits a “simple" arithmetical interpretation, but > space, dimension, energy remains in the shadow. > > Science has not yet decided between Plato's and Aristotle’s conception of > reality, and not all people are aware of the hypothetical nature of those > options, nor that Digital Mechanism, or even just the Church-Turing thesis, > leads us back to Pythagorus and Plato. > > The natural numbers realised the infinities without making them into existing > things, or beings. > > Bruno > > > > Also, some may think that because theoretical physics (field theories) is > expressed in a language that includes a mathematically continuous (real > number) background R^4 of spacetime and the methods of multidimensional > calculus (tensor calculus, etc.), that because mathematically > infinite-divisibility is present and infinitary definitions (like "limit") > are present, that these "infinities" of the mathematics are real in the > actual world.
I agree. Infinities are useful does not entails that infinities exist in “Reality”. That is why I can be more open to your fictionalism for second order arithmetic, analysis, set theory, more than for arithmetic (that would make Mechanism and even just computer science into fiction).. Bruno > > @philipthrift > > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/8728448c-eab1-4e12-86ee-4e3b765b9237%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/8728448c-eab1-4e12-86ee-4e3b765b9237%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/02D5A42D-574D-46AC-94C9-02CC80061D0C%40ulb.ac.be.
