Mastodawn

1/7
This air-quote #Mathematics unquote article https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/ keeps appearing in my feed, and I initially made some comments the first time, debunking it from a #Maths point-of-view, but given how it keeps popping up I think I need to do a more thorough #MathsMonday thread about it

Firstly, the author is a Physics journo, so you can take what he says about #math with a grain of salt (for some reason I see a lot of them doing this overreach, instead of checking with a Mathematician)…

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Dec 1

2/7
"the strange infinite ones that other mathematicians ignore" - Mathematicians "ignore" them because there's no such thing

"Set theory deals with the infinite" - no it doesn't. Sets are finite, literally have an end. They have cardinality - the number of elements in the set. You can't have a cardinality of "infinity", nor is it a set if there's no closing bracket (which there can't be if it's infinite)…

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🇺🇦 haxadecimal 🚫👑Feb 27

@SmartmanApps
I'm not a mathematician, but in multiple mathematics courses I've taken, set theory has been taught including both finite and infinite sets.
If you don't believe Wikipedia, Wolfram MathWorld also discussed both finite and infinite sets.
As you say, an infinite set cannot be defined by enumerating its elements. However, that is not the only way to define a set.
There exist, for instance, infinite sets of the natural numbers, integers, rationals, algebraic numbers, reals, etc.

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Feb 27

@brouhaha
"I'm not a mathematician" - I'm a Maths teacher.

"infinite sets. If you don't believe Wikipedia" - you won't find infinite sets in Maths textbooks. Stop looking at Wikipedia for Maths - it can't even get order of operations correct! 😂

"infinite sets of the natural numbers" - they're not a set, they're just "the Natural numbers".

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Active Mouse Feb 27

@SmartmanApps @brouhaha just chiming in here since I just followed haxadecimal, but there are plenty of textbooks that mention infinite sets.

This even starts implicitly at an early age - here's an example of a textbook which, on page 46, says that the solution to an inequality is "the set of all real numbers x that make the inequality true." So the solution to x < 1 is the set of all real numbers less than 1, and there are certainly infinitely many of those!

https://www.scribd.com/document/536766882/389788543-Edexcel-Pure-Maths-Year-1

But pick your favourite textbook on Group Theory, say - Jordan & Jordan is a good one. On page 41 they give examples of groups (which they define as a certain type of set) are the complex numbers, real numbers, natural numbers and other infinite sets.

Of course, any textbook on set theory will include infinite sets :) E.g. Halmos, Jech, ...

But let us call these things something other than sets. What Cantor showed remains true, whatever you call them: there is a bijection between the natural numbers and the rational numbers, but not between the natural numbers and the real numbers. I assume you don't disagree with that... or do you?

Edexcel Pure Maths Year 1 | PDF

Scribd is the world's largest social reading and publishing site.

Scribd

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🇺🇦 haxadecimal 🚫👑

@ActiveMouse
Apparently SmartmanApps is a finitist. They typically believe that natural numbers exist, but don't consider them to be a set. That lets them have their cake and eat it too, acknowledging that there is no greatest natural number, but defining infinite sets away by claiming that it is not valid to construct a set that cannot be enumerated in full.

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🇺🇦 haxadecimal 🚫👑Feb 27

@ActiveMouse
Nothing we can say possibly convince a finitist that infinite sets exist. It is effectively a religious belief. Infinite sets offend their sensibilities, so they declare them not to exist.
He also seems to believe that finitism is something accepted by more than an extremely miniscule number of mathematicians.

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Active Mouse Feb 27

@brouhaha yes, in my (not overly long, I should say, but long enough) mathematical career I don't think I've ever met an actual finitist. Certainly not one who admitted to it, compared to a great many evident non-finitists. I'm trying to remember how much analysis you can do with actual finitary mathematics, I had an idea it ran out before the intermediate value theorem? In reverse mathematics you need some fragment of second order arithmetic to prove it, meaning a concept of a set of natural numbers but no higher order types or larger cardinalities.

They aren't talking like a finitist though. A finitist would, I think, recognise that they're talking about a philosophical position rather than a mathematical one and grant that no inconsistency has been found in, say, RCA_0, rather than bluntly denying proofs. Indeed, a finitist would agree that ZF- (or some weaker set theory) proves Cantor's theorem, because the theory and the proof can be coded in PA and verified there. a finitist wouldn't say the proof is wrong, instead they'd say it's not talking about anything real. Thing is, I'm not a Platonist so I agree, I just don't think the natural numbers themselves are "real" either.

I see from their other comments linked from their pinned post that they have other misconceptions about mathematics. It would be interesting to see what their views on those really are but they don't seem to be engaging here!

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Mar 2

@brouhaha @[email protected]
"Apparently SmartmanApps is a finitist" - never said anything of the sort

"don't consider them to be a set" - they're not

"not valid to construct a set that cannot be enumerated" - hence we use intervals to describe them, see worksheet

"religious belief" - infinite sets is, yes, it's literally impossible to fit an infinite amount of anything into a finite space

"more than an extremely miniscule number of mathematicians" - as is evident from textbooks using intervals