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NNTP-Posting-Date: Sun, 23 Jun 2024 16:05:41 +0000
Subject: Re: it's a conceptual zoo out there
Newsgroups: sci.math
References: <v57u7g$rgs$1@dont-email.me> <v594kk$bco0$1@dont-email.me>
 <v59ffh$d4vt$1@dont-email.me>
From: Mike Terry <news.dead.person.stones@darjeeling.plus.com>
Date: Sun, 23 Jun 2024 17:05:42 +0100
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On 23/06/2024 16:37, sobriquet wrote:
> Op 23/06/2024 om 14:32 schreef FromTheRafters:
>> sobriquet pretended :
>>> In particle physics, people used to refer to the particle zoo since there was such a bewildering 
>>> variety of elementary particles that were being discovered in the previous century.
>>> Eventually things got reduced to a relatively small set of fundamental fermions and bosons and 
>>> all other particles (like hadrons or mesons) were composed from these constituents (the standard 
>>> model of particle physics).
>>>
>>> Can we expect something similar to happen eventually in math, given
>>> that there is a bewildering variety of concepts in math (like number, function, relation, field, 
>>> ring, set, geometry, topology, algebra, group, graph, category, tensor, sheaf, bundle, scheme, 
>>> variety, etc..).
>>>
>>> https://www.youtube.com/watch?v=KiI8OnlBTKs
>>>
>>> Can we kind of distinguish between mathematical reality and mathematical fantasy or is this 
>>> distinction only applicable to an empirical science like physics or biology (like evolution vs 
>>> intelligent design)?
>>
>> I don't think so because regarding physics there is one goal, to model reality, and I believe only 
>> one reality to deal with. With mathematics there are endless abstractions such as the idea of 
>> endlessness itself in its many forms.
> 
> I think there is still a general trend towards unification in both math and science.
> In both cases things get discovered and explored and when things are
> explored in more detail, often connections are discovered between seemingly unrelated fields that 
> allow one to come up with a unified framework that underlies things that initially seemed unrelated.
> 
> https://www.youtube.com/watch?v=DxCWRAT0WKc
> 

What does happen is that lecturers teach their material to students year upon year upon year, and 
over time the ideas and methods are distilled to become more efficient from a teaching perspective. 
Theorems that were once long and complicated are approached in a more efficient way, and the proofs 
may turn out to be quite short.  Often the shortness hides a wealth of smaller results, but still 
there is a big improvement in understandability, and the connections between areas become better 
understood.

I doubt all the above would be unified into /just/ one concept, because they reflect different 
interests in what is being studied.  That doesn't mean they won't be seen as aspects of some simpler 
ideas - for example when I studied maths all the above were seen as sets.  However that didn't mean 
there was just one course (on set theory) that covered all the above - even though you might say 
"aha - everything is just a set, so that's it."  (At that time category threory was a bit too new to 
base the entire degree on, but I imagine these days category theory provides a similar (better?) 
unififying view of the various areas, like set theory in my study days.  But still there are 
numbers, topologies, sets, manifolds, rings etc.).



Mike.