Advice

39 views Jun 19, 2023

Modern mathematics is seriously confused when it comes to "infinite sets", "infinite processes", "limits" and "real numbers". Where does this befuddlement come from? A lot of it has to do with not being precise enough about the distinction between a description of a mathematical object and a specification of that object. The latter ought to be completely faithful to the actual definition of the object in question: the former is much more ambiguous, open-ended and overused. If you want to make progress in exploring mathematics, being mindful of this crucial distinction is very valuable.

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hello everyone I'm Norman warburger

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today I want to talk a little bit about

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the important distinction between

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describing mathematical objects and

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specifying them

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so this is an important distinction that

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plays a big role in trying to understand

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the difficulties with real numbers and

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an infinite set there and indeed a lot

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of modern foundational approaches to

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pure mathematics so about a hundred

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years ago when Cantor introduced his

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thinking especially about infinite sense

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there was vigorous debate about the

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merits of such in fact most

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mathematicians opposed it they could see

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through it but as time went by for

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various reasons the debate subsided

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but the difficulties didn't really go

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away so these days most students are

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happily unaware of The Logical problems

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with uncomputable numbers or

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non-measurable functions or the Axiom of

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choice or hierarchies of Cardinals and

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ordinals or various anomalies or

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paradoxes that supposedly appear in

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topology set theory analysis Etc

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in my view the paradoxes or anomalies

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aren't simply consequence sort of

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obvious consequences of not thinking

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clearly about foundational issues right

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if you've got a proper theory of math

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facts there are no paradoxes people okay

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it's a it's a litmus test it's it's

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telling you in your face that you are

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not thinking correctly

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so to to really understand the

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difficulties it's good to go back to the

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basic division in mathematics between

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discrete and continuous so there's a

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world of difference in terms of

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complexity level between these two

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aspects of mathematics

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discrete mathematics you know starting

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with the natural numbers integers

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fractions very concrete and and relying

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on Elementary counting

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continuous mathematics which really

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comes from the physics and the

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measurement process

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gets easily involved in

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talking about so-called irrational

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numbers like root two and root 5 and Pi

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Etc and um and that's a source of a lot

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of the difficulties in the modern uh

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Theory so

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up to 100 years ago the notion of the

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Continuum appeared intuitively

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straightforward perhaps but it was

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acknowledged to be difficult to pin down

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precisely but with the Advent of quantum

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mechanics and uh 20th century physics

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the concept grew increasingly murky

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because if space has an inherent

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graininess and is not infinitely

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divisible and time is perhaps relative

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and perhaps fine at an extent then what

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exactly are we modeling with our our

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notion of the number line so while

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engineers and scientists today view real

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numbers primarily as infinite decimals

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mathematicians rather regard them as

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equivalence classes of koshy sequences

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or perhaps dedicated cuts and each of

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these views has serious difficulties

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they're a little bit different but sort

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of corresponding but always there is

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this crucial problem of discussing

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infinite objects without sufficient

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regard to how to specify them okay so

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there's an implicit assumption that you

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can just use the words oh this is an

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infinite object or we're considering an

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infinite set Etc without

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the responsibility of saying precisely

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what such a thing is and specifying such

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a thing

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so let me illustrate this difference so

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consider a finite sequence

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say the sequence l

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sequence one

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five nine so this sequence can be

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described in quite a lot of different

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ways for example as the increasing

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sequence of possible last digits in an

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odd integer square or perhaps as the

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sequence of numbers less than 10 which

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are congruent to one modular four or as

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the sequence of digits occurring that

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246th prime number after you you remove

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repetitions so there's lots of ways of

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describing it but ultimately there's

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only one way to specify the list

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completely and unambiguously namely you

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have to explicitly list all the elements

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one five nine all those other

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descriptions

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are secondary to the actual

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specification where you actually lay the

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the elements out in their entirety

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completely so everybody can see

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and here you can you can already

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understand why there is such a big

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difficulty in this jump to supposedly

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infinite sequences

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so for example suppose we consider the

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infinite sequence M which is

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three

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five

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seven

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dot dot dot

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okay now there are people to whom that

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means something it doesn't mean anything

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to me because the dot dot dots are you

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know haven't been specified

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you can see here that the situation is

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dramatically different because it's not

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possible to explicitly list all the

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elements of an infinite sequence it's

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impossible so you're going to have to

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shift to describing a rule for

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generating such a sequence

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but this rule

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is typically highly non-unique

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for example in the case I just showed

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you m could be the list of all odd

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numbers starting with

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three

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or maybe it's the list of all odd Primes

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or there's countless other possible

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descriptions that start three five seven

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but without such a rule a definition

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like 357. is really meaningless so the

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right model to have in mind perhaps when

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modeling an infinite sequence is a

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computer okay which is churning out

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number after number on to some long tape

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and the process never stops

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as long as you keep supplying more tape

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and electricity and additional memory

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banks but the sequence is not to be

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identified by the completed output tape

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that doesn't mean anything okay it's

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it's a figment of our imagination

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what we actually have is a computer

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program that's just running and

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generating more and more digits or more

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and more output onto this tape that's

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the thing which is concrete and

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completely specifiable

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however then you have to deal with this

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enormous problem of

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saying when two different computer

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programs aren't going to generate this

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same sequence

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and that's a famous unsolved probably

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unsolvable problem in computer science

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so we should be loathe to you know

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accept that as a foundational aspect of

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building up mathematics if we have to

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rely on computer science and there's

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this major unsolved problem which is

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you're intimately connected with what

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we're trying to do then we have a

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problem too

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what happens when we go to sets from

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sequences so a finite set like the set

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two four six eight can also be described

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in many ways but ultimately it too is

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only properly specified by showing all

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its elements

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okay in this case order is not important

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because sets are not ordered so they the

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elements could be scattered over a page

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for example

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it sets

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cannot be so explicitly exhibited

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should be considered in a in a secondary

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kind of fashion they should be viewed as

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not having been properly really

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exhibited so an example might be suppose

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we consider the set of all odd perfect

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numbers less than 10 to the 100 to the

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Thousand now we don't know if there are

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any odd perfect numbers we do not know

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if there are any odd perfect numbers

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less than 10 to the 100 to the thousand

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so I can write down this this s is this

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set but that's really a description okay

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and that description does not deserve to

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be called a specification of the set at

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least not with our current understanding

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of perfect numbers because we can't even

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decide whether this set s is empty or

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not

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okay so it's not a specification it's

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just something much much weaker it's

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like a weak description

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so with sets the dichotomy between

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finite and infinite is much more severe

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than for sequences

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because we do not allow a steady

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exhibition of the elements through time

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okay it's impossible to exhibit all the

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elements of an infinite set at once so

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the notion is an ideal one that more

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properly belongs to philosophy it can

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only be approximated within mathematics

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the notion of a completed infinite set

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is actually contrary to classical

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thinking

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since we can't actually collect together

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more than a finite number of elements

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as a completed totality why pretend that

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we can

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it's the same reason that God or the

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Hereafter are not

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generally regarded as proper scientific

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entities both infinite sets God and the

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Hereafter may very well exist in our

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universe but this is a philosophical or

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religious inquiry not a mathematical or

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a scientific one

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so I know that there's all kinds of

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reasons why we would like to believe in

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infinite sads and infinite processes

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being finished because it makes a lot of

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complicated things simpler it's a it's a

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massive simplification of the

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mathematical world that's that's what

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we're currently living in in modern peer

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mathematics an enormous simplification

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of something which is actually much

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richer and more complicated than it than

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it appears to be to an undergraduate

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the idea of infinity as an unattainable

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ideal that can only be approached by an

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endless sequence of better and better

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finite approximations is both humble and

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ancient and one that I would strongly

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advise to those wishing to really

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understand mathematics more deeply

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this is the position that our committees

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Newton Euler Gauss would have taken and

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it's a view that really ought to be

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seriously

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resurrected and placed center stage in

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terms of a new and Bolder and more

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logical mathematics

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it's time for us to grow up