On Monday, July 13, 2020 at 7:07:36 AM UTC-5, Bruno Marchal wrote: > > > On 10 Jul 2020, at 20:57, Lawrence Crowell <[email protected] > <javascript:>> wrote: > > On Friday, July 10, 2020 at 7:21:29 AM UTC-5, Bruno Marchal wrote: >> >> >> On 7 Jul 2020, at 16:10, Lawrence Crowell <[email protected]> >> wrote: >> >> On Monday, July 6, 2020 at 9:29:31 AM UTC-5, Bruno Marchal wrote: >>> >>> >>> On 4 Jul 2020, at 19:25, Lawrence Crowell <[email protected]> >>> wrote: >>> >>> On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote: >>>> >>>> >>>> On 3 Jul 2020, at 14:30, Lawrence Crowell <[email protected]> >>>> wrote: >>>> >>>> It does not need that sort of intense structuring. The extension of >>>> real to complex numbers is a case of forcing of real numbers into pairs >>>> that obey multiplication rules for complex numbers. >>>> >>>> >>>> >>>> That is not forcing in the theoretical meaning of the papers referred >>>> to by P. Thrift. >>>> >>> >>> It is and is not. Agreed forcing is a dense set theoretic technique on >>> how to extend a model into some other model. I think it was Hamkins who >>> wrote on how forcing was really not that mysterious and used the example of >>> extending integers into rationals, those into reals and those into ,,, . >>> Set theory without some connection to actual mathematics is about as >>> useless as tits on a boar hog. >>> >>> >>> >> I think mathematics is best when it can be used to actually calculate >> things, whether that be with the physical world or in expanding areas of >> mathematics itself. The one thing that always caused some distaste for >> axiomatic set theory is that it often seems so detached and remote from >> anything else. >> >> >> >> Set theory is useful to make clear the semantic of simpler system. Cantor >> was trying to make sense of the differential equation of heat propagation. >> >> But, unlike arithmetic, there is no real consensus of what a set should >> be, and there are different theories, like ZF and NF. >> Intuitionist set theory can have application in numerical analysis, >> constructive computer science, etc. >> >> Then, there is that extraordinary use of the most abstract (and >> theological) part of set theory (the Large large cardinal) which have been >> used to discover and proof the existence of some interesting order >> structure on the (infinite) braids, … >> >> We never know. We cannot predict if some mathematics will or will not >> been applied. Hardy apologised for doing useless mathematics (number >> theory) which today is used each time we pay something on the net… >> >> Bruno >> >> > That is a part of my understanding. We have ZF set theory, with and > without the axiom of choice, and then there are other very different set > theories, such as Polish set theory. Then there is the HoTT homotopy type > theory. It seems the goal of set theory to reduce all of mathematics to > some completely fundamental objects, whether sets or types etc, that we > have a plethora of these systems and a host of complexity. > > > > We know today that there are no foundational theory which could based all > of mathematics, and it is important to continue the distinction between > basic concepts like numbers and functions from their representations in set > theory, or even in category theory. But will mechanism, we get a very > simple ontology to explain the “physical experience”, without having the > assume a physical reality. With or without Mechanism, the fact that > universal machine exist makes it impossible to have any complete theory > about reality, whatever that reality is. But that means that even just in > arithmetic, we must expect an infinity of surprises in the exploration. > With Mechanism, *we* are such surprises already. > > Set theory should be renamed. It is really the theory of the infinite(s). > It is a sort of vertical theology: how to figure out what looks like a > universe of sets. It attracts the lover of vertigo … Then, like always in > mathematics, unexpected application rise up, like the apparent link between > braids and some *very* large cardinal (Wooden, Laver, …). > > Bruno > > > I prefer in ways category theory, for at least it is connected to topology with Grothendieck and Etale cohomology. I am not highly learned in the subject, just the basic essentials. There is also the Langlands program where all theory can be reduced to numbers. The monstrous moonshine of Borscherds is interesting, and the bosonic string obeys the automorphism on the Fischer-Greiss group.
We may never come to any pinnacle on this. It is my suspicion the human condition can be summed up as two stone ages punctuated by a disequilibrium situation. We happen to be in that disequilibrium situation, we call it civilization and history. The first stone age was in the Pleistocene when the Earth was biologically rich and the next one is in what might be called the anthropocene where the Earth will be depleted in a post-mass extinction period. I have suspicions we are near the peak of this disequilibrium situation, and if we are to come to some sort of summation of things, or at least known be a few of us, it might have to happen pretty soon. LC -- 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/9ba547d4-8e49-4375-890c-c92d7417b9f7o%40googlegroups.com.
