Daniel Rubin
Daniel Rubin
861 subscribers
What sense can we make of the part of mathematics that admits constructions we can never write down or compute?
I take up this question in defense of infinite processes and analysis with NJ Wildberger, professor of mathematics at the University of New South Wales and host of the Youtube channel Insights Into Mathematics, https://www.youtube.com/channel/UCXl0...
Timestamps:
0:00 Intro
3:13 NJ outlines his issue with real numbers and infinite sequences
6:55 Exact vs. approximate; the equation x3 = 15.
17:44 Statement of my position. When pure mathematics diverged from applied mathematics.
23:25 My first objection: we can make sense of the constructions in real analysis.
27:34 Objects defined by choice or by algorithm
32:46 What constrains what we can speak of in mathematics?
34:30 My explanation of Cauchy sequences in terms of measurement
37:05 Solving polynomial equations computationally. Power series as formal objects or as functions.
40:53 My second objection: analysis is a very successful theory
48:10 Is there a crisis that necessitates a rejection of infinities?
54:06 Do we have to get the foundations right before proceeding? The sum pi + e + sqrt(2).
1:01:40 My third objection: what is computable?
Norman's online course Algebraic Calculus One: https://www.openlearning.com/courses/...
Check out my other podcast episodes for interesting discussions of math and other topics: https://www.youtube.com/playlist?list...
And check out my Tricky Parts of Calculus playlist https://www.youtube.com/playlist?list... for careful exposition of the subtle and difficult parts of calculus, including treatments of pi, e, and other topics mentioned in this video.
00:00
all right norman i'm so excited thanks
00:03
so much for coming on my show
00:04
let me tell you and uh you know but uh i
00:07
tell my audience who you are
00:09
why you're on my show today um
00:12
no you are norman wildberger you are a
00:15
professor of mathematics
00:16
at the university of new south wales uh
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now retired right
00:21
recently retired that's right okay and
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you have this wonderful youtube channel
00:26
insights into mathematics how long have
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you had this
00:30
youtube channel by the way oh that's
00:31
been quite a while um probably
00:33
close to 15 years now 15 years so yeah
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it's
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sort of towards the beginning of youtube
00:39
it must have been
00:40
yeah not too long afterwards yeah uh
00:44
so i discovered this channel a few years
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ago uh while i was a postdoc and i was
00:48
doing some teaching
00:49
of my own and i i wondered what was
00:52
there on youtube and i discovered your
00:53
you have wonderful lectures
00:55
on on this channel especially
00:57
emphasizing
00:58
uh the historical development of the
01:00
subjects which is something that
01:02
you don't see in many places so i really
01:05
like
01:05
i guess your episodes on galwa theory
01:08
some episodes on some ancient
01:10
babylonian mathematics something that's
01:12
of interest to me as well
01:14
um and you also now have this uh open
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learning course
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uh algebraic calculus which i did check
01:21
out
01:22
uh i i checked it out i um
01:26
because it's algebraic which is uh very
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odd to me
01:29
as as an analyst where i i think
01:31
calculus is about the theory of
01:33
functions of a real variable i was
01:35
checking out how you handle certain
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things
01:37
uh but there's a lot of interesting
01:39
content there very different from a
01:40
regular
01:41
calculus class um well the thing about
01:44
your channel though so you have this
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wonderful
01:46
you have this stuff i really like and
01:48
you have a lot of stuff i was uh
01:49
very um well i was surprised to find uh
01:54
elsewhere on your channel because you
01:55
have some rather unorthodox views
01:58
of mathematics uh about analysis
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in particular the the area of math in
02:04
which i work
02:05
and uh i was very surprised find you
02:08
have a very strange perspective
02:10
which we are going to try to uh
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battle about i guess or at least come
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try to come to some kind of
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understanding about
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and see if uh see if people can
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understand your position and see if i
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can just
02:23
destroy it as much as i uh respect you
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well we'll just see what
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what happens uh is that is that uh
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meetable to you
02:30
yeah look that all sounds that's all
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sounds really good daniel um
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i think it's a great uh to have
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discussions about foundational issues in
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mathematics i think we need
02:39
so much more of that um there is a lot
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of
02:44
things to do at that foundational level
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it's not just
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all you know that these stratospheric
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really complicated questions are the
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only things that really can engage young
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people
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i think it's really important to
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recognize that there's potential for
02:58
moving into really well established
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fields and and seeing them in a new
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light by asking you know what's
03:04
really going on here then does this
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really work
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um and i agree with that i agree with
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that that scent
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yeah at least so and and yes it's
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certainly true that i have uh
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non-orthodox views i i
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have quite different points of view on
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on the nature
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of of the continuum on on the role of
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real numbers in
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in the the role of infinity i i have
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different points of view as to
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um how close to come computer
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or computer you know uh issues
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mathematics
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should be um i would much rather that we
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move more
03:41
closer to what's going on in the
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computational world
03:45
and i think that's that's really a place
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where we can anchor our mathematics and
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not have it
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be what it is currently which is is
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almost a branch of philosophy really uh
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the foundations of modern mathematics
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are really uh largely philosophical
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unfortunately
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so i even i even might right i even
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might agree with with that uh on
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on some level so okay so i'm an analyst
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i uh i use i use the real numbers i use
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the continuum all the time i've taught
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classes about
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about the real numbers what is can you
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try to summarize
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what is your issue with
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uh analysis well okay so let me just
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take that statement of yours that you
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made so confidently you know that you
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use real numbers
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i would suggest that the truth is
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actually something a little bit
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different
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that you use the term real number okay
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but what the when you actually use are
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floating point numbers
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that is well or unless you're doing
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exact mathematics where you can work in
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the realm of of rational numbers and
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finite decimals and everything is
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completely cut and dried and there's no
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ambiguity whatsoever and all
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computations
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finish but if you're not working in that
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framework
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then you're probably working in what's
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really historically an applied
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framework you know the mathematician
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the mathematics of engineering and
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physics which
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which concerns itself with with
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approximate quantities with measurements
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which are
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determined up to a certain level of of
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resolution
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and correspond to you know decimals of a
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certain
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length and often there's a bit of
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indeterminacy you know how much
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resolution do we have how much accuracy
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do we have at some point
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now the real number framework which
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ostensibly is the foundation of
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mathematics
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purports to to claim that we're working
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with with infinite
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decimals with with with um with numbers
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whose whose decimal expansions go on
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forever and essentially encode an
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infinite amount of information and that
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that our entire numerical system is
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really ultimately based on these
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these objects which have themselves uh
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infinite amounts of
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information in them and you know that to
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me is
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is almost absurd and and then clearly
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not
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what what happens computationally you
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know obviously
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so we're kind of playing a game where
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we're pretending that we're
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that we're we we have a theory of real
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numbers with infinite
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decimals and that we're able to to
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perform arithmetic with these things
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you know and we and we use the language
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as if we were able to do that but the
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reality is is very very different the
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reality is that
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basically we have to do the same thing
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the applied mathematicians
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do we have to make computations up to a
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certain decimal point and then after
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that
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we don't know what's going on or or
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there's a computer program that can keep
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on going but
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it's not going to finish
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i see uh so you say when i um
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say when i introduce uh or think of
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things in terms of the
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koshi sequences you're saying there's
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something
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uh i'm glossing over i i think i know
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what i'm talking about
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but uh i'm i'm doing something that i
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that i shouldn't because uh there's some
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there's some
07:14
there's some infinity there what what is
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it uh
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yeah look i i can tell you what what it
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is uh
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in terms of an example which i think is
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probably useful to sort of anchor the
07:27
discussion so
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let's consider a really important
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distinction
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is between the exact and the approximate
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you know there are some things in life
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which are exact and
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and exactly specifiable and you and you
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can
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resolve them exactly and then there are
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other things in life which are not that
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way
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you know and i guess in real life we we
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recognize that most things are kind of
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approximate and very few things are
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truly exact maybe mathematics is almost
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characterized by being you know that
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domain in which exactness
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can arise so anyway let's let's keep
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that distinction in mind the distinction
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between exact
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and approximate and that's
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um that's a really important distinction
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so let's let's consider two equations
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let's consider the equation say 3x
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equals 15.
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elementary equation that's
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solvable exactly we use arithmetic we
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find that x equals 5.
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finished that's the answer it's exact
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we're all going to get the same answer
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now consider a slightly more complicated
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equation which is
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x cubed equals 15.
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x cubed equals 15. okay now
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that is a lot a completely different
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equation
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and the the reason is because that
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equation actually has
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no exact solution
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now that's not something that my fellow
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mathematicians are going to say but that
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is actually the computational reality
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the reality is that
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i have to put x that satisfies
09:00
x cubed equals 15.
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but but of course there's lots of
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approximate ones you you get your
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calculator you get your computer you ask
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it for the cubed root of 15. it will
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give you a decimal expansion
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it happens to be 2.4 i have here
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2.46621207 because i computed it earlier
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because i was going to use this example
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but that's not the exact answer that
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that's that's only the beginning of an
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algorithm that you could
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keep on running you could keep your your
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computer going and get
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generate more and more digits but at no
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point you actually stop and get a an
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exact output
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at any point even thousands of years
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from now you're only going to have an
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approximate
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answer to this question of what is the
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solution of x cubed equals 15.
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that's just intrinsically an approximate
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question
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okay and and us pure mathematicians we
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would like to pretend
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we would like to believe that there is a
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number
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which satisfies this thing miraculously
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and so we give it a name we generate
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some symbolics we
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we create this theory of real numbers
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that supposedly supports
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things like this but it's
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computationally
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uh unrealistic it doesn't correspond
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with what
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computer programs do
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hmm uh
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i have okay i you you're going in a
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direction i didn't think you would go in
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because i have a
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my objection to that example is somewhat
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different from maybe your objective
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objective to say um if you were going to
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talk about
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what is there really a circumference to
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a circle or something because
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uh you know that there is i think i feel
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like you're choosing a certain formalism
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where where you can't have an answer but
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we but we could
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for for in a nice equation uh
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like that like x cubed equals 15
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that is different from the issues that
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would arise in
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in maybe some other places in analysis
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if you if you know what i mean
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um i guess uh
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maybe i i i will want to
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pin down um what it is
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exactly that you are objecting to
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about i mean
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okay you've had me going in some
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different directions than i expected
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because
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when you talk about i agree with you
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about this issue about exactness and
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approximation
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and i feel that somehow the the
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formalism of real numbers
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exactly addresses somehow in their
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definition they are
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they are objects meant to capture the
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uh approximating spirits so in which so
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that limits can exist so that the kind
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of point that you might want on the
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uh on the graph where the two curves the
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cubic and
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and the line uh y equals 15
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must they must intersect at at some
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point
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that is that that is somehow the point
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of that formalism
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and yes that's that's right that we we
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would like in in our in our sort of our
12:13
landscape we would like to
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take what the applied mathematicians are
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doing
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and and and the sort of approximate
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nature of the
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objects and maybe the functions that
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they're dealing with the the procedures
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we would like to sort of bring it into
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into an exact and precise framework
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where
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where everything is is is exact
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and that's and that's what the real
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number game tries to do so it tries to
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create this
12:39
this sort of system which is uh beyond
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the world of computation
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okay it's beyond the world of
12:45
computation it's up there in some
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stratospheric
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level of our own you know mental
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thoughts
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which we can then you know say aha
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that's where that's where these things
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live um we're not obligated
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to to to talk in the language of the
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computer scientists we have our
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our our own special world you know in
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which
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we are able to get our solutions and in
13:08
which we interpret the
13:10
the the things that we write down
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and this is this is to me a religious
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like a religious fantasy
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it's but it's even in some sense even
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more
13:20
absurd than a lot of religious fantasies
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it's um it's completely
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disconnected from from computational
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reality
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and it's really motivated by wishful
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thinking massive amounts of wishful
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thinking because we would like
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we would like there to be a cubed root
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of 50 we would like there to be a
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fundamental theorem of algebra
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we would like that our our definitions
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of integration and so on actually make
13:43
sense
13:44
just real quick you know i mean i feel
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like we could easily
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uh have a formula where we adjoined you
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know cube groups of
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cube roots of 15 and we wouldn't be
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plagued with some of the
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uh the issue you know they don't have to
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live within
13:58
the rational numbers or say but we could
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know how they work how they
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how we could compute with them right
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so that's that's an important point to
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address so
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certainly yes one can create algebraic
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frameworks in which something like
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the cube group a cubed root of 15 exists
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or the square root of minus one
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or the square root of minus one or other
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things like this and
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um you know mathematicians algebraists
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who are not analysts are able to do this
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and
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they can do this without any infinite
14:32
processes
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right and and to to the audience who's
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not familiar with that i mean
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probably the nicest way to of thinking
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about it is in terms of embedding
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our current arithmetic in in the
14:44
somewhat larger world of matrices
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so once you embed our our you know
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familiar sort of rational numbers
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in in as diagonals of matrices then you
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can
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use this larger matrix uh to to create
14:58
things that like that have the property
15:00
that x cubed equals 15.
15:02
but crucially those objects that you
15:04
construct
15:05
have no direct interpretation as a point
15:09
on the number line somewhere between 2.4
15:12
and 2.5 that's that's no longer you
15:15
you've gone down a completely separate
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path
15:20
now when i have discussions about this
15:22
with people on the internet
15:24
often these two points of view are
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confused and people think that because
15:27
there's an
15:28
algebraic construction of some
15:31
extension field in which you can yes you
15:34
can create an object which has the
15:35
property that it's cubed root that it's
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cube is 15. that this somehow justifies
15:40
the the claim that that we know what the
15:42
decimal 2.46621207
15:46
etc actually means but that's that's
15:50
very wrong thinking they're actually
15:51
very separate things and you have to
15:54
choose which which direction you're
15:55
going to go in
15:56
you know and if you're working in the
15:59
decimal framework then
16:00
you're basically destined to follow the
16:02
computer scientists and work in an
16:04
approximate setting
16:07
if you work in the algebraic setting
16:09
with extension feels well then things
16:11
get
16:11
there's other attendant difficulties due
16:13
to just the the algebraic complexity of
16:16
things
16:17
and the fact that that one extension
16:19
field doesn't interact very
16:20
nicely necessarily with another
16:22
extension field
16:24
you have to work harder to to have an
16:26
arithmetic in which there's both a cubed
16:28
root of uh
16:29
15 and a square root of minus one or or
16:32
whatever
16:34
so all right i just thought it was
16:35
algebraically a lot more complicated
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true so you would see currently
16:40
currently we have this this this
16:42
religious idea that
16:43
all the complexity is in the plane
16:46
there's just the plane of complex
16:47
numbers
16:48
that's all people you know and all all
16:52
algebraic equations have solutions
16:54
somewhere here
16:56
this is it's it's really a medieval uh
16:59
philosophy that
17:00
the the best and the best uh analog that
17:03
i can think of is
17:04
is the um the medieval thought that that
17:07
the
17:07
the night sky was captured by the idea
17:10
of the celestial sphere
17:12
it's just this two-dimensional sphere
17:14
around us
17:16
we're in the center this is
17:17
two-dimensional sphere and all the stars
17:19
are sort of attached to this sphere the
17:20
sphere rotates around us in the night
17:22
sky
17:24
right and you know in the sun and the
17:26
moon and all the planets are sort of
17:27
attached to this two-sphere
17:28
it's all relatively uh simple and and
17:31
you know
17:32
it's not that simple but it's relatively
17:34
simple
17:35
yeah but then yeah the motion of the
17:37
planets changed all that
17:39
they had to have a crisis to realize
17:42
they weren't in that
17:42
in that framework and i i guess i want
17:45
to
17:46
challenge all these things that that you
17:47
are saying i i'm not often called
17:49
religious
17:50
i've been called really only a few times
17:53
in my life
17:54
uh but i i do i do want okay so let me
17:57
just say uh the way i
17:59
might what my position is because i i
18:01
don't think that everything that you're
18:02
saying is is
18:03
is total nonsense i mean i do think that
18:06
the distinction that you
18:08
want to make about uh some kind of
18:11
computability
18:12
some kind of deal the way that we
18:15
compute or actually uh
18:17
i guess because we deal with computers
18:18
today uh algorithmically some finiteness
18:21
uh to to what we're doing is an
18:23
important distinction i'm not dismissing
18:25
it out of hand
18:27
uh i think there can be a lot of great
18:29
mathematics that takes that as
18:31
uh it's its foundation as as its
18:34
starting point
18:35
uh if you ask questions within that
18:37
framework okay so i don't actually want
18:39
to dismiss
18:39
any of the things that come from that uh
18:43
but i don't see a reason to reject
18:45
analysis or to
18:46
to view it as uh this as a religion
18:49
uh relationship i don't want to reject
18:51
analysis what i want to do is i want to
18:53
build
18:55
from a more solid foundation see
18:58
analysis is
18:59
is um is it has been for for you know
19:02
most of its history an applied subject
19:04
whose concerns have been with
19:06
with real life um problems with
19:09
describing physical motion with
19:11
describing
19:12
uh waves with sound uh with trajectories
19:16
etc
19:17
okay so that's that's that historically
19:20
that's what analysis has been
19:22
only relatively recently maybe in the
19:24
last 100 150 years has this
19:27
sort of pure mathematics development
19:29
started
19:30
you know pure mathematics is not
19:32
historically uh
19:34
really anywhere as old as applied
19:37
mathematics
19:38
and so this this pure attempt is attempt
19:40
to try to
19:41
to try to take what the applied
19:43
mathematicians are doing and model it
19:45
in in sort of an abstract way um
19:48
independent of of physics or or or
19:52
other uh you know direct um
19:56
aspects of the world can you say where
19:57
can you say when to the new thing
19:59
can you say when you think uh things
20:01
changed or what development was so
20:03
important
20:04
yeah i would say it looks somewhere
20:05
around the middle of the 19th century
20:08
um you know um there's aspects of chaos
20:12
work uh george boole
20:14
maybe you have some claim to being one
20:16
of the first
20:17
really pure mathematicians um
20:20
with the birth of with a birth of of of
20:22
uh
20:24
of hyperbolic geometry and sort of the
20:26
realization that that that
20:27
space had um wasn't intrinsically canty
20:31
like there wasn't there wasn't just one
20:33
space there could be different spaces
20:35
you know and it was a question of you
20:37
know how we define
20:38
particular objects that creates a space
20:41
um with the realization that algebraic
20:43
structures had
20:44
had more flexibility that you could you
20:46
could um you could think about uh
20:48
groups or fields or rings or vector
20:51
spaces sort of in an abstract setting
20:53
that they didn't have to be you know the
20:56
the familiar things
20:59
um so sometime in the 19th century i
21:01
think this development
21:02
started it occurred to people that
21:04
there's sort of this this abstract world
21:06
of of mathematical objects that could be
21:08
created and defined and studied sort of
21:10
independent of of the
21:12
real world of of you know physical
21:15
interactions etc but i guess i've never
21:17
understood your um
21:19
your issue as being with abstraction so
21:22
much as it is
21:22
with um infinite processes and and
21:26
computability oh yeah
21:27
i have nothing to do with abstraction i
21:28
mean applied mathematicians have also
21:30
been interested in abstraction i mean an
21:32
applied mathematician is
21:34
engaged in an interesting problem
21:35
they're naturally going to be thinking
21:37
but uh what's
21:37
you know what's in that sort of abstract
21:39
point of view on this
21:41
um so that's that's not it i
21:44
uh i'm not and i i think you know pure
21:46
mathematics is is is a wonderful
21:49
uh initiative um it's just that we have
21:52
not
21:52
we have not really recognized that it
21:54
really involves us
21:56
taking what the applied mathematicians
21:59
have done and saying okay we now have to
22:02
have to
22:02
find uh an independent way of of setting
22:06
all this
22:07
up we can't just assume that that that
22:10
we have an underlying number system for
22:12
example
22:13
you know that that obeys the properties
22:16
that that it
22:16
seems like like a continuum has
22:21
you know we have to actually if we want
22:22
a number system we have to build it
22:24
we have to define it we have to we have
22:27
to define what the operations are we
22:29
have to prove
22:30
that that the laws of arithmetic that we
22:32
want are actually valid
22:34
we we can't just you know adopt that and
22:36
say well that's what happens in in the
22:38
real world and
22:39
let's just pretend that there's a real
22:40
number system which satisfies
22:42
properties a b c d up to a h
22:46
which is which is by the way what a lot
22:48
of even calculus and analysis textbooks
22:50
ultimately do when you actually look at
22:52
you know a text on analysis and try to
22:54
see where are they
22:56
setting up a theory of real numbers it's
22:58
mostly vacuous not always but but mostly
23:01
it's true that some some do give you
23:04
know just try to state definitions of
23:06
of what a real number system is or what
23:08
what properties it would have to satisfy
23:10
some do construct it
23:12
in in the manner at least you're
23:13
suggesting but i'm sure you have some
23:14
objection to
23:16
uh using these there's dedicant cuts or
23:20
koshi sequences which is the way i i
23:22
think of real numbers
23:24
um i guess so i have i have some
23:27
objections to the the view that you're
23:30
putting out
23:30
um and i i really had to think about
23:33
this because
23:34
uh you know you make a strong case and i
23:37
it this goes against things that are
23:39
very deeply ingrained in my
23:41
thinking as a mathematician of course of
23:43
course of course
23:44
uh so here's uh here's a way of thinking
23:47
about that i mean
23:48
right it's easy to to end up in a
23:50
philosophical conversation about these
23:52
things
23:53
you know where you know in my opinion
23:55
your opinion and so on
23:56
i think a way of trying to anchor this
23:58
and to try to remove the philosophical
24:01
aspect is to
24:02
is to concentrate on what we can
24:04
actually write down
24:06
you know not what we can what we can
24:08
imagine or what we can imagine that we
24:10
can imagine
24:11
but what we could actually write down
24:14
and and
24:15
and demonstrate you know in an explicit
24:17
fashion so that anybody can come along
24:19
and
24:19
check you know what we're talking about
24:21
and and and verify that what we're
24:23
saying is true
24:25
well that is that that is that i want to
24:26
talk about you want to talk about
24:28
prime numbers which which are bigger
24:30
than a trillion to the trillion
24:32
okay you know i can say let's consider
24:34
the set of prime numbers which are
24:35
bigger than a trillion to the trillion
24:38
yeah um you know um maybe
24:41
mankind knows a few such but not not too
24:44
many a handful of them
24:45
okay but if i i could i can still say
24:47
well let's consider the set of all such
24:48
prime numbers
24:49
but that's not something that i can
24:51
write down in any reasonable fashion i
24:53
can't i can't say here
24:54
here's the set that i'm that i'm
24:56
thinking about it's a purely almost a
24:58
purely fictitious thing and whether
25:00
it really goes on forever or not is is
25:02
highly questionable
25:04
and is in you know it wrapped up with
25:06
all kinds of philosophical questions as
25:08
to the nature of of space-time and the
25:10
extent of the universe
25:11
etc etc right acceptance
25:14
we can avoid all that by just saying
25:16
let's let's be computational
25:19
let's open up a computer program and and
25:21
and write down stuff that we can encode
25:23
in
25:23
in mathematica you know and and
25:26
everything else will be philosophy
25:29
well but i i have to say that strikes me
25:31
as a kind of philosophical
25:33
position to to start with i i i guess
25:36
you know my my first my objections come
25:39
in i guess three headings and that the
25:41
first one
25:42
is is to attack that in my view
25:45
these infinite processes or
25:48
possibly not checkable by a finite
25:53
algorithm these are nevertheless things
25:56
that if they come from
25:58
the kinds of constructions that go
26:00
between you know
26:01
that use the full set of natural numbers
26:04
or or
26:05
involve the continuum are nevertheless
26:07
comprehensible
26:08
or well-defined in some sense and that
26:10
that's what we need as mathematicians so
26:11
i guess
26:12
like like in a philosophical sense well
26:15
i
26:16
i guess i don't i am not myself a set
26:18
theorist
26:19
or a logician uh i don't concentrate on
26:23
these sorts of uh foundational issues
26:27
in in quite the same way some people do
26:29
but but maybe they would
26:30
there is some issue of consistency of uh
26:34
you know this uh axiom of infinity or or
26:37
something like that
26:38
uh or i am perfectly happy for example
26:42
to
26:43
uh discuss you know i i feel like i know
26:46
what is a
26:47
sequence uh of a fun a sequence on the
26:50
define on the natural numbers
26:52
say to the rational numbers without
26:54
without having to have
26:56
already a a process to
26:59
by which to check whether two sequences
27:02
possibly specified in different ways
27:04
uh are actually the same or if they
27:08
or if they have a certain limit um as
27:11
long as that
27:11
definition is well defined it may be in
27:15
the sense of sets that
27:16
the equality is some equivalence
27:18
relation
27:19
that that's uh that's my first class of
27:23
objection uh relates to the what i guess
27:26
i would call the comprehensibility
27:27
of these these constructions which you
27:30
you want to say
27:31
are in our imagination okay can i ask
27:34
you a question about your understanding
27:36
of sequences
27:37
um this is a a point that that i've
27:40
talked quite a bit about in my my
27:42
youtube series the distinction between
27:44
computable sequences and and what i call
27:47
choice
27:47
sequences so there's a bit of a
27:49
dichotomy in analysis you know whether
27:52
you know if you're talking about a
27:53
sequence say of natural numbers
27:55
sequence of natural numbers are you talk
27:57
are you implicitly assuming that that
27:59
sequence
28:01
is generated by a computer program that
28:04
has some computational finite basis
28:07
or are you allowing the possibility that
28:10
this
28:10
sequence is generated by you know some
28:14
perhaps process outside a universe that
28:16
that's able to
28:17
to to invoke a an infinite amount of
28:21
computing power
28:23
yeah i guess i'm uh implicitly referring
28:26
i never thought about this but i am
28:28
referring
28:29
maybe to these uh axioms of set theory
28:32
which say what sorts of constructions
28:34
you can make with sets
28:36
and uh once you know that there is this
28:38
set uh the natural numbers
28:40
uh you know and functions are these sets
28:43
of the subsets of the product set of two
28:46
sets
28:47
yeah so in all sorts of considerations i
28:48
think a really a good first thing to
28:51
hone
28:51
your your thinking is to at least decide
28:54
whether your notion of infinite sequence
28:57
is is on the computational side or
28:59
whether it's on the
29:00
on the choice side and those are very
29:03
very different positions so when someone
29:05
says i believe in sequences
29:06
that's the first thing that i want to
29:08
ask them if
29:10
a computer scientist is probably not too
29:13
happy
29:13
thinking about a choice sequence they
29:14
want to restrict themselves to computer
29:17
programs you know because that's you
29:18
know i here's a program and it generates
29:19
a sequence
29:20
and but but pure mathematicians
29:23
especially those who've been sort of you
29:25
know trained
29:26
in in in axiomatic set theory they know
29:28
about this
29:29
this this um your axiom of choice and
29:32
and
29:33
and and for that you kind of need to
29:34
have an orientation where you're
29:36
you're kind of willing to allow the
29:39
contemplation of sequences which are
29:42
generated by oracles
29:43
outside our universe that don't require
29:46
a finite amount of time
29:48
time to to or or energy to specify
29:51
but those are very two different those
29:52
are very two different uh you know
29:54
points of view to this and
29:56
and your your interaction with a lot of
30:00
theorems of analysis are going to be
30:02
different if you have these two
30:03
different orientations
30:04
so that's like a really a good first
30:06
starting point is to decide
30:08
which side of the fence if if at all
30:12
of this you're going to be on
30:15
yeah i guess i don't i don't need my
30:17
sequences to be specified by
30:20
any particular kind of algorithm
30:22
satisfying any any property
30:24
i feel like um part of what i mean by a
30:26
comprehensible
30:28
i mean i i think people once they know
30:32
there's some number of integers then
30:34
there's always the next one
30:36
they feel comfortable talking about the
30:38
the set of all of them
30:39
and there's an idea well if they're uh
30:42
i can imagine assigning any you know a
30:45
rational number to each integer
30:48
that's you know part of the construction
30:49
when we start talking about sequences
30:51
the kinds of sequences
30:52
out of which we build the real numbers
30:56
well you know that's that's that's fine
30:59
um
31:00
to a certain extent i mean but it
31:02
doesn't it doesn't really correspond
31:04
with what computers do like
31:05
if you look at a computer program when a
31:08
computer program
31:09
is um is oriented towards identifying
31:12
the type of an expression
31:13
you you input something on your computer
31:16
and then you press the enter button you
31:17
know and your computer will look at what
31:19
you've typed and will try to decide what
31:20
it is that it you know it will maybe
31:22
recognize oh you he's just input a
31:23
natural number
31:25
or he's input a rational number he's
31:26
input a set or he's inputted a list or a
31:29
string or something or other right
31:30
so the computer the computer does not
31:32
have a memory location where it has all
31:35
the natural numbers stored
31:37
you know and if you put in a natural
31:38
number then it it searches to see if it
31:40
matches with one of the natural numbers
31:42
on its list
31:44
now rather it has a a procedure to
31:46
determine
31:47
whether an expression has a given type
31:49
or not
31:51
and for that the computer did so that
31:53
that
31:54
avoids the requirement that the computer
31:56
has this huge
31:57
memory uh you know of all of all of all
32:00
natural numbers
32:01
so this idea that that there is a set of
32:04
natural numbers
32:05
and that this set is is a legitimate
32:07
mathematical object that's not shared by
32:09
our computers
32:11
now our computers recognize that if i
32:13
keep on
32:14
typing in numbers eventually i'm going
32:16
to flow over the the size of the memory
32:18
and that the computer's going to say
32:20
stop i can't deal with it anymore
32:22
you know um and that's the nature even
32:26
of our own world i mean if you make
32:28
computations eventually the numbers get
32:30
so big that they overwhelm you
32:32
you you you you quickly can get outside
32:35
of uh
32:35
the realm of uh you know
32:39
okay i think i'm back in business here
32:41
okay so what were we
32:42
started we were discussing um
32:46
so i i guess i want to know uh
32:49
maybe this gets into the philosophy you
32:50
wanted to avoid but
32:52
uh it's sort of central to this whole
32:54
issue why restrict
32:56
to what is uh computable i
32:59
i wonder if you've viewed your your uh
33:02
this restriction which which again i i
33:04
don't
33:05
view i don't dismiss it as something
33:07
worth thinking about
33:09
but you know there are other people who
33:10
will say for example well we shouldn't
33:12
talk about numbers that are too big
33:14
because
33:15
in the universe there's only so many
33:17
atoms you know there
33:18
if you think there's finally many atoms
33:20
then why why should you think that there
33:22
is a number bigger
33:23
than whatever its atoms or other
33:25
subatomic particles
33:26
do you view your objection as as being
33:29
somewhat
33:29
analogous to that or in some other way
33:33
i guess i i see it as being like um
33:37
analogous to a scientist wanting to uh
33:41
restrict themselves to things that can
33:43
be observed
33:46
you know um that's maybe the science has
33:48
to has to find boundaries so that you
33:50
know um
33:52
mystical things are not getting embedded
33:56
in science and one way of doing that is
33:58
let's let's restrict ourselves to things
33:59
that we can actually observe and we can
34:01
make
34:02
um you know we can we can compare
34:05
observations and
34:06
the corresponding thing for mathematics
34:08
is what you can write down and what you
34:09
can compute
34:11
um that's that's to me
34:14
to me a kind of analogous uh kind of
34:16
thing i i wonder because
34:18
there are some situations where there
34:20
there are things that are
34:22
true but they're they're hard to verify
34:25
there are
34:25
uh situations where where we do i
34:29
don't i guess we use this and you know
34:30
when i think of a koshy sequence
34:32
i i'm uh to me it's like a very real
34:35
thing like what the
34:37
a scientist does when they try to
34:38
measure something i mean maybe i'll
34:40
explain
34:41
when you what is the length you know of
34:43
of some
34:44
object you measure it with a ruler and
34:47
the ruler is ruled however fine you know
34:49
maybe in millimeters and you say well
34:51
here's a here's a measurement
34:53
7.6 centimeters or something
34:56
but no one imagines that the thing is
34:58
actually 76 of some
35:00
fundamental quantity they they know that
35:02
the root that ruler comes with a
35:04
precision
35:05
right uh and there's the idea well if
35:08
you held up a magnifying glass
35:10
and you took a ruler made much finer
35:13
right maybe you would add another uh
35:15
decimal place
35:16
of of precision you know according to
35:19
that that next one
35:21
and you could do it again with a
35:22
microscope and then the idea of a koshi
35:24
sequence is
35:25
imagine you could do that indefinitely
35:29
yeah but that it's that indefinitely um
35:32
and the use of the word
35:33
imagine and indefinitely that i object
35:35
to
35:36
let's imagine that we can do something
35:37
indefinitely i mean you might as well be
35:39
talking about
35:40
you know let's imagine that i can jump
35:42
to andromeda
35:43
or let's imagine that i am god or or you
35:46
know i mean there's all kinds of
35:47
mystical things we could be talking
35:48
about that don't actually correspond to
35:50
what you actually
35:51
can do so i want to restrict ourselves
35:54
to a discussion about
35:55
what we can actually do not but not what
35:58
we could imagine that we might be able
36:00
to do if we were completely different
36:01
creatures from what we
36:02
are i see but i i feel i guess this this
36:05
does affect another philosophical view
36:07
of
36:07
what we do as mathematicians in my view
36:10
uh
36:10
we create abstractions abstract models
36:15
things that obey rules that we that we
36:17
put forward
36:18
in order to try to model like some some
36:20
real phenomena
36:21
something that's in front of our face
36:23
that we want to get at but people wanted
36:24
to get at
36:25
the the hypotenuse of an isosceles right
36:28
triangle or the uh
36:29
you know the circumference of a circle
36:32
and you know you
36:33
you do make these inventions and you
36:35
know they are abstract in some sense
36:37
uh yeah you cannot actually keep going
36:40
with
36:41
better and better and finer rulers but
36:44
you know once you if you have some sense
36:48
of what that is maybe
36:49
i'm just talking about well-definedness
36:51
in in terms of sets
36:53
and you can deal with those properties
36:55
as you have put them forward
36:57
then you can have a meaningful theory
37:00
behind that that's
37:01
that is my perspective on why why these
37:03
things are comprehensible
37:05
well let me give you an example of
37:07
something that i've been thinking about
37:08
um
37:08
recently so in the last month or or so
37:12
i've been um
37:13
writing making some videos on solving
37:16
polynomial equations
37:18
um this is on my sister channel called
37:20
wild egg math so i'm i'm proposing to
37:22
solve polynomial equations famous
37:24
question of how you solve a quintic
37:26
equation or a more general polynomial
37:27
equation
37:28
and i have a sort of a power series
37:30
point of view so this is you know this
37:32
is out in the spirit of
37:34
of a power series that that keeps on
37:36
going
37:37
and i'm looking for solutions which are
37:39
of power series formant
37:41
and um we were able to you know find
37:43
some really attractive formulas that
37:46
connect beautifully with with a whole
37:48
branch of combinatorics
37:50
around the catalan story and and
37:52
subdivisions of polygons and so on
37:55
and and one of the things that's
37:57
involved there is
37:58
you know to try to interpret what do we
38:00
actually mean if we have
38:02
a power series you know are we really
38:05
are we really thinking about something
38:06
that's infinite that keeps on going
38:08
are we thinking about a function see
38:12
the the re the analyst wants to think of
38:13
a power series as a function
38:15
as something which inputs a a specific
38:18
number and outputs a specific number
38:21
but because there is this infinite
38:22
number of terms that that requires
38:25
the arithmetic to support being able to
38:27
do an infinite number of things
38:30
and that's that's why you sort of need
38:31
the theory of real numbers to be able to
38:34
you know get an output to a power series
38:37
now the combinatorialists however have
38:40
have a very different way of thinking
38:41
about this
38:42
so combinatorialists you think of power
38:44
series as as generating functions
38:47
as like close hangers on on which you
38:49
you you put combinatorial objects
38:51
and it's not necessarily important to
38:54
to talk about the identification of you
38:57
know what what
38:58
what you get when you when you when you
38:59
put x equal to some number
39:02
it's a power series it's it's it's a
39:04
formal object
39:05
and you can deal with that and that
39:07
that's what i've been doing
39:08
with this polynomial uh equation story
39:12
you can deal with that sort of one level
39:13
at a time like you could say well if we
39:15
go up to
39:17
terms of degree 10 you know and then we
39:20
have some identity that we want to prove
39:23
the way we interpret that is not in
39:24
terms of of sticking in real numbers and
39:26
evaluating what we interpret that is
39:28
if you if you truncate all things up to
39:31
a certain level say a level 10
39:33
then the both sides of the equation are
39:37
are verifiable and that this hole is not
39:40
just for 10 but it also holds for 20 and
39:41
for 100 in fact for any value
39:43
so if you can prove some some uh some
39:46
family of
39:47
of identities like that then that gives
39:49
you an interpretation of what the power
39:52
series is
39:52
meaning without having to interpret it
39:55
as a real valued function
39:57
which in my view doesn't make any sense
40:00
yeah so you see that there's
40:01
there's once you once you have some
40:03
skepticism towards real numbers
40:05
you are naturally led to to look for
40:08
alternate ways of interpreting and
40:10
thinking about
40:11
what's really going on and arguably
40:15
that can open up all kinds of new doors
40:17
because you know now i think
40:19
we have we have solutions of polynomial
40:21
equations
40:22
more clearly laid out than we did before
40:25
and
40:26
and and a lot of beautiful combinatorics
40:28
ends up uh
40:29
becoming involved well there okay i
40:32
don't know your
40:33
uh the the combinatorics that you're
40:35
talking about i am aware of
40:36
ways of of solving uh the fifth degree
40:39
and and higher
40:40
polynomials uh either with theta
40:43
functions or
40:43
the modular function um or you know just
40:47
newton's method uh it might also you
40:49
might want to apply that
40:51
i guess um well maybe this my second
40:55
heading of of objection we're sort of
40:57
going there anyway
40:59
uh because we're uh at odds about
41:02
whether we can accept this model is what
41:04
about the fact that i feel like analysis
41:06
as i understand it with with limits and
41:09
infinite processes is very successful as
41:11
a theory
41:12
uh would would you agree with that i
41:14
mean it gives us
41:16
it look it was already very successful
41:18
in the time of euler
41:19
i mean um you know euler was uh was i
41:22
was a master and by that time already
41:24
the
41:25
the 18th century mathematicians had
41:27
already
41:28
um applied calculus analysis to solve a
41:31
whole wide variety of problems
41:34
right um you know and and that did not
41:36
require
41:37
all this limited gadgetry uh which um
41:40
you know ended up being
41:41
sort of tacked on to it in the latter
41:44
part of the 19th century
41:45
well what about oil oiler was quite
41:48
happy to uh
41:49
to to work essentially algebraically but
41:52
with a certain certain amount of freedom
41:54
um and solve all kinds of of practical
41:57
problems
41:58
that's right until until until 48 came
42:00
along
42:01
until which until fourier and fourier
42:04
series i think was the the thing that uh
42:07
necessitated all this uh looking at the
42:10
the foundations of
42:11
uh of calculus and that's right that's
42:14
right that
42:15
that was a really important point
42:17
because it was first of all
42:18
non-intuitive
42:20
and also uh people in fact objected to
42:23
the same kinds of things that i'm
42:24
objecting that that for you implicitly
42:26
was saying well we need to add up these
42:28
infinite numbers of functions and
42:30
and if we do this in some some way then
42:32
then we get these sort of remarkable
42:34
uh unlikely uh you know identities
42:37
well but do you do you re reject these
42:40
uh these identity because i'm i'm trying
42:42
to say that
42:42
oh no i mean i i don't read i i i think
42:45
they again they should be interpreted
42:47
in in an approximate setting this
42:50
there's a natural
42:51
way of interpreting these things which
42:53
is that um you know if you take a
42:54
fourier series and you go up to terms of
42:57
degree 10 or degree 20 or whatever it is
42:59
that you want
43:00
then you're going to get some some
43:02
something which is which is
43:03
perfectly reasonable and that's not
43:05
going to be exactly the thing that that
43:07
solves your equation but it's going to
43:09
be close enough to
43:10
to solve whatever problem that you're
43:12
interested in
43:13
i mean i suppose to me the the the the
43:16
fourier theory is
43:17
is not a final thing it's not as if you
43:20
you you put up all
43:21
you put all the terms together and you
43:23
get some final object
43:25
to me it's a process that you can go
43:27
down you can get your computer to run
43:29
and whatever and then you can stop at
43:31
some point and calculate well how far
43:33
are you from some ideal solution that
43:35
has some property you want
43:38
yeah that's ultimately more or less sort
43:40
of what computers do anyway you know
43:42
that's sort of
43:43
because they can't calculate the entire
43:45
fourier series so
43:46
you know well so that that leads me to
43:48
think of two different things
43:50
one is that maybe we're saying almost
43:53
the same thing but
43:54
we're we're choosing different languages
43:55
because a lot of
43:57
what's built into analysis and the real
43:59
numbers or
44:00
anything else is when you take a limit
44:02
it's well after you get a certain
44:04
you know to a certain point you you are
44:06
this close and so there's a lot of that
44:08
approximation
44:10
built in and we're just using a language
44:12
and a formalism that says
44:14
well in in in the limit this this is
44:17
identically equal to the thing
44:19
but it's not just a formulation you're
44:20
actually positing
44:22
a huge um step beyond what we can do
44:26
you know uh i you're positing that that
44:29
we we can go beyond
44:30
the universe to to some some distant
44:32
universe and talk about the res the
44:34
final result of this
44:36
infinite amount of of of computation
44:38
yeah i i think we can we can find the
44:40
button at the back of our laptop
44:42
and and and and and press the go to
44:45
infinity button that will
44:46
that will that will generate the entire
44:50
you know algorithm and and finish it and
44:52
give us an output
44:54
yeah i have a formula when i put the
44:57
wrong
44:57
it just this just doesn't correspond
44:59
with what really is
45:01
i yeah you're right that i when i when
45:02
we put all the elements on that closed
45:04
line whether it's a power series or a
45:06
fourier series and i
45:07
i have a theorem from within the
45:09
definitions of this of this theory and
45:11
that says it
45:12
yeah when all summed up it is equal to
45:14
that that's what i choose
45:15
i guess you could say that's what i
45:16
choose to believe but somehow different
45:19
from a religion with the
45:20
faith and prayers and whatever i'm
45:22
trying to say it's not
45:23
too religious it's not too different
45:25
from my point of view uh-huh well
45:27
it is uh sorry it's not too different
45:29
from religion
45:31
different from a religion i i see it as
45:34
as
45:34
as just a mathematical religion i see
45:37
which people like to believe in because
45:39
it because it coincides with what they
45:41
want to believe in
45:43
but what about what about the fact that
45:44
this is why i try to say
45:46
i view this theory as very successful i
45:48
i this is a theory that
45:49
you know it explains are these these
45:52
irrational numbers and lengths it also
45:55
solves
45:56
differential equations uh look at those
45:58
all those things all those things
46:01
were already in place prime uh mostly
46:04
already before dedicated uh
46:07
cuts and and wire straws limits came in
46:10
laplace oiler lagrange these great
46:14
18th century mathematicians could
46:16
already solve a lot of these problems
46:18
and modern day physicists also don't
46:21
need
46:21
don't need the limit theory they're
46:23
quite capable of just
46:24
using their their power series uh
46:27
approximations and generating
46:29
as many terms as they need to in order
46:31
to get answers
46:32
so i think it's fundamentally wrong to
46:35
attribute the success
46:36
of of applied analysis
46:39
to the theoretical developments that
46:41
were latched on to it
46:43
after the fact after the
46:46
well so how about after that because in
46:48
my view like buyer strass
46:50
came long after euler
46:53
yes uh
46:56
uh they are not intrinsically necessary
46:58
for actual applications
47:01
hmm they are they are an icing on the
47:04
cake they are
47:05
a kind of addressing uh uh you know that
47:08
that
47:08
allows people to think okay now we have
47:10
a theory that kind of makes sense
47:14
in my view a lot of numerical analysis
47:18
does it derives its inspiration from
47:21
you know first you have the analytical
47:22
results which which have nothing to do
47:24
with
47:25
computers but they give you a process
47:28
that you could then
47:29
put into a computer that that's at least
47:30
the paradigm i'm i'm operating
47:33
uh you know i guess um what what i know
47:36
from
47:36
my my work in this field of
47:38
approximation theory
47:40
it is all it is it combines this uh
47:43
analysis of you know well what function
47:46
is closest to what
47:47
in this space is closest to this
47:49
abstract thing
47:51
uh and and it uses a lot of these
47:53
analytical results
47:55
uh with differential equations or or
47:59
what have you a lot of this formalism
48:01
but it does produce in the end the kind
48:03
of thing that you want to compute it's
48:04
the key
48:05
science for scientific computing and i
48:08
guess
48:08
i i don't see that um the reason i try
48:12
to say that we have
48:13
a very successful theory and i
48:16
want to know whether you acknowledge
48:18
that is because i usually these changes
48:20
um
48:21
when you demand a change in our
48:23
foundations or the way we look at
48:25
something
48:26
uh it's usually precipitated by some
48:29
kind of a
48:30
crisis in the sense of uh thomas kuhn
48:33
are you familiar with uh i i guess um
48:38
i wonder if we really have that that
48:41
kind of crisis
48:42
a disagreement something that is uh
48:45
we cannot separate it's not just
48:48
different language and
48:49
you know oh we we can view it as look at
48:52
this as just
48:52
meaning approximations uh versus here's
48:55
a language where i view well
48:57
when you make the full thing it's it's
48:58
the whole thing do we really
49:00
disagree on on something
49:03
is is what i'm saying i'm
49:06
i i absolutely agree that the the role
49:09
of approximation
49:10
is crucial um that you know how much
49:13
error there is
49:14
after you've done a certain uh operation
49:16
this this is an essential a question
49:19
right and and to me this is this is
49:22
different from
49:23
the theoretical idea that you know that
49:25
we can go to infinity and that
49:26
there's an exact answer it's exactly
49:29
because
49:29
we can't go to infinity and we can't
49:31
obtain an exact answer
49:33
that we are forced to deal with the
49:35
issue of approximation
49:37
and how close are we and how how good
49:40
are our estimates
49:44
oh certainly that's as an analyst that
49:45
that to me is that the heart of what
49:47
doing mathematics is about
49:49
uh you know bounding some some new
49:51
quantity you don't
49:52
know exactly in terms of things you that
49:55
you know better
49:56
that's uh getting these estimates but of
49:59
course i'm constantly
50:00
referring to you know functions defined
50:02
that are themselves defined
50:04
via some limiting process which is
50:06
always happening whether you have a
50:07
differential or
50:08
equation or or anything else really a
50:10
variational problem
50:12
uh i'm constantly performing these
50:15
operations on
50:16
you know the real functions or
50:19
uh you know things that use this this
50:22
formalism
50:23
okay but it's not much of a shift to
50:25
instead of thinking about functions
50:27
which are defined in terms of limiting
50:28
process
50:29
to think in terms of approximate
50:31
functions
50:34
which are not given by limits but which
50:36
are given by algorithms which you can
50:38
decide where
50:39
when you're going to truncate or stop
50:41
the algorithms
50:43
that and that corresponds with what
50:45
computers are i think
50:46
more essentially really doing so i think
50:49
we're
50:49
sort of saying the same things here yeah
50:52
computers i guess uh i still don't know
50:56
why computers have to figure into
50:58
what we can what we can even comprehend
51:00
mathematically or abstractly
51:02
uh that was in my first point but i i
51:04
guess if
51:06
if we have my point now is if we have a
51:09
theory
51:10
that you know its practitioners believe
51:12
that they understand
51:13
and it's and it's fantastically
51:15
successful in doing what we wanted to do
51:17
which is
51:18
describe the real world or provide us
51:19
with a theory that's full of um
51:22
lots of interesting behavior but the
51:23
ability to say something about it
51:26
why should we reject it and and make it
51:29
you know oh we should view it as
51:30
actually undo that language actually
51:33
everything needs to be thought of as the
51:35
approximation
51:35
and look back at the rational numbers
51:37
and and how they uh
51:39
you know get this close to the thing
51:41
that we want uh
51:42
we should throw out the language of you
51:45
know
51:46
the real number the finished thing they
51:48
think they continue
51:49
so i would say that i would want to
51:51
restate that a little bit is that
51:52
we've we've had a lot of success and
51:56
i mean applied mathematicians have been
51:58
very successful
51:59
okay um they've had a long history
52:02
they've done all kinds of really amazing
52:04
things and now we are at the stage
52:07
that we are trying to develop pure
52:09
mathematical frameworks
52:11
that accurately capture all these
52:14
successes that the applied
52:15
mathematicians have been able to do
52:18
but i i'm in my view we have been too
52:21
quick
52:22
to try to create these frameworks and
52:24
haven't spent enough time on on the
52:26
foundational issues
52:27
and as a result our understanding of
52:30
what's really going on
52:31
is often a skew so in my view what we
52:33
have to do is we have to take these
52:35
admitted successes of the applied
52:37
mathematicians and the applied analysts
52:39
and we have to look at them in a
52:43
different way in in in a computationally
52:45
explicit
52:47
and careful way that we can so that we
52:49
so everything can be on the page and
52:51
then we have a
52:52
a really coherent story of what's really
52:54
going on there
52:55
so to me to me pure mathematics is in
52:58
some sense a kind of a
53:00
relatively new discipline in in the
53:02
large scale history
53:03
of mathematics it's uh it's an emerging
53:07
discipline
53:08
which has a lot of potential uh as
53:11
we as we you know look through all these
53:13
various applied things
53:14
and try to figure out what's really
53:16
going on here and you know and my view
53:19
is that is that uh orienting oneself
53:21
with computation
53:22
and and distancing oneself from from
53:24
infinite
53:25
uh processes and and and other
53:28
unrealities
53:29
is the way to go it's it's so funny that
53:31
you say a lot of your
53:32
language is the some of the same
53:34
language that i use but but in a
53:36
different
53:36
debate with uh 20th century mathematics
53:40
and where i feel it went astray with
53:42
uh trying to write frameworks
53:45
like the category theory and a lot of
53:48
growth index mathematics
53:50
which i i think is is leading
53:52
mathematics astray but but somehow in a
53:54
different direction from what
53:56
we're talking about and i view analysis
53:59
as very concrete
54:01
even if it's not all computable in the
54:03
sense that you want
54:05
i don't agree about um your emphasis on
54:09
on
54:10
foundations uh i wonder what you think
54:12
about this
54:13
in my view it's very hard to know maybe
54:15
what are the found
54:16
foundational what what's the right you
54:18
know you know the people
54:20
who tried to write down these
54:21
definitions of a set had a lot of
54:23
trouble and had to uh
54:24
try to work it out many times um
54:28
and and it's still an unfinished project
54:31
uh but that's never bothered me you know
54:33
i could do a lot of mathematics
54:35
understanding what i understood of you
54:37
know the natural numbers and the real
54:39
numbers
54:40
and i never had to think about uh
54:43
the continuum hypothesis uh
54:46
or or higher infinities and i just feel
54:49
like it's
54:49
very hard to worry about those things
54:51
unl until
54:52
there's an issue that really brings it
54:55
to the to the fore
54:56
uh that that makes that forces us to to
55:00
make a decision
55:01
about it somehow and i i guess i don't
55:03
see what what is the instigating event
55:05
if we really differed on you know
55:08
a theorem and you think it's this way i
55:11
think it's the other way
55:12
and it's not just a matter of language
55:15
then we'd really know that we
55:17
where are disagreements i mean i have
55:19
lots of examples like that that i that
55:20
i'm throwing out to my audience all the
55:22
time
55:22
there's the one standard one which which
55:24
any of my long-term viewers will hear me
55:27
say many many times which is you know um
55:31
i i claim that even even elementary
55:33
arithmetic with real numbers
55:34
is is essentially taught logical or
55:36
vacuous and my standard example is to
55:38
calculate
55:39
um one-half plus one-third plus
55:41
one-fifth
55:43
no actually that's that's the rational
55:44
number version that that one we can do
55:46
the the real number version is what is
55:48
the sum of
55:50
pi plus e plus square root of two
55:54
and i think you know that that's like
55:58
that's like the real number
55:59
analog of asking a grade six student
56:02
what is one half plus one third plus one
56:03
fifth is arguably
56:04
sort of the simplest uh irrational
56:06
numbers
56:08
and you know invariably you don't get a
56:11
response no one is able to calculate
56:13
this
56:14
so so how can it be that you have an
56:16
arithmetical structure
56:17
if i take the three simplest irrational
56:19
numbers in your arithmetical structure
56:21
and ask you to do the simplest
56:22
operation on them namely addition that
56:25
that that
56:26
you just get mute you just get no one
56:28
can give me a proper answer
56:31
it's the answer is pi plus e plus square
56:33
root of two the answer is the same thing
56:35
as the question
56:36
that is a nice name for that number and
56:38
you can know what the kinds of things
56:40
you want okay
56:41
this just demonstrates to me completely
56:43
clearly that
56:44
it's a tautological arithmetic i mean if
56:46
you can't even do
56:47
you know you can sum up the three
56:49
simplest irrationals then
56:51
well i don't know what other names
56:54
i don't know if this is really in the
56:56
main thread of our discussion but i of
56:57
course i would say
56:58
it's i've heard you on your channel say
57:00
many times and i i mean i agree that um
57:02
the infinite decimal formalism
57:04
for real numbers you know is not it does
57:08
not clearly
57:09
uh specify you know what what is the
57:11
meaning of doing the operation of
57:13
addition or multiplication
57:14
infinite decimals are good for order uh
57:17
but of course think of the infinite
57:18
decimal as one representation of the
57:20
koshi sequence
57:22
koshi sequences have the property that
57:24
it's cl that the sum of two koshi
57:26
sequences is a koshi sequence and so the
57:27
meaning of
57:28
the arithmetic is that you would uh you
57:32
know
57:32
sum individually in in the nth term
57:35
the nth rational number element of the
57:38
koshi sequence
57:38
and that of course the theory tells you
57:40
that that is uh
57:42
you can prove that's another koshy
57:43
sequence so it must be another real
57:45
number i mean
57:46
does that does that answer your
57:47
questions question right now no no of
57:49
course not i mean i i want to i want an
57:51
answer like if somebody if i asked
57:53
a primary school student what's one-half
57:55
plus one-third plus one-fifth i would
57:57
uh eventually hope that they figure out
57:58
that it's 31 over 30.
58:00
you know and and that's that's the
58:01
answer so i want an answer like that
58:03
you know that's that's somehow a
58:04
different i mean sometimes like when you
58:06
solve a differential equation you don't
58:08
get
58:08
you know what the name of that function
58:10
is you know it's
58:12
it's that it it does this it's not it's
58:13
not equal to
58:15
yeah but we're not talking about
58:16
differential equation we're talking
58:17
about the simplest
58:19
operation on arguably this the three
58:21
simplest irrational numbers you know
58:24
it's not some some some rocket science
58:26
kind of arithmetical question but if you
58:28
wanted to compare it to the size of any
58:31
other real number you could by virtue of
58:33
how well you understood what pi
58:35
e and root two were right how how well
58:38
you could approximate them with rational
58:39
numbers
58:40
uh that would that would be how you'd
58:42
how you do it if you wanted to ask other
58:44
questions about that number like uh you
58:47
know
58:48
it's uh whether it was yeah but that's
58:50
assuming that there are such numbers you
58:52
know that's assuming that
58:53
you know okay that there already is such
58:57
a framework in which this entire
58:58
discussion is taking place
59:00
i'm just using this as a question for
59:02
you
59:03
to to to think about whether you
59:04
actually are are
59:06
confident that this framework actually
59:08
really makes sense
59:10
like to me it doesn't make sense i i
59:11
don't believe that there's a real number
59:13
called pi
59:13
i don't think there's a real number
59:14
called e i don't think there's a real
59:16
number
59:16
called square root of two and i don't
59:18
think there's a well-defined addition of
59:20
real numbers
59:21
so you know
59:24
i i guess i i don't know should i
59:25
usually explore that like do you do you
59:27
think um
59:28
you know you can take this uh uh one
59:31
plus
59:32
one over n to the n and there's a
59:34
sequence
59:35
it's a nice sequence but uh you know it
59:37
has
59:38
these nice properties you know you can
59:40
prove that it is
59:41
increasing and it's bounded
59:45
you know and i i i can prove to you that
59:47
uh i have a construction base of the
59:49
coaching sequence every every uh every
59:52
set of real numbers with that property
59:54
is going to have a limit
59:56
you know and you can you can get as
59:58
close to it as as you like
60:00
uh maybe that sequence converges rather
60:02
slowly there are ones that converge a
60:04
lot
60:04
faster that that are that are nice for
60:07
computability purposes
60:08
right and that that's that is the way i
60:10
i know real numbers and it's the same
60:12
way that i know
60:13
uh even even those more complicated
60:15
things like solutions to differential
60:16
equations they're almost always
60:18
constructed
60:19
as some kind of a limit
60:22
right and i have a theory to say when
60:24
those limits exist and that that somehow
60:26
you know being able to appeal to that is
60:28
like the final step
60:30
uh where and i use all those practical
60:32
estimates based on the approximations in
60:34
order to get that final step but that's
60:35
that's our that's our language that's
60:37
how
60:38
that's how the system works
60:41
i i but i guess i wondered if there was
60:43
some instigating
60:45
event that caused you to it but
60:48
so to you it's like if this even this
60:50
arithmetic
60:51
uh with these numbers there isn't a
60:52
better name for that for that number but
60:54
i feel like that the kinds of things
60:56
that you'd want to know about that
60:57
number
60:58
using to me that number is when you say
61:01
that number i have it's
61:03
there's quotes in my mind you might as
61:04
well talk about that leprechaun or
61:06
that that loch ness monster if you took
61:08
an if you took another number
61:10
uh real number with uh representation
61:12
but i don't i don't believe in real
61:14
numbers
61:14
right you know you took any you took
61:16
anything you're making is is one that's
61:18
incomprehensible
61:19
you could take any rational number and i
61:21
could tell you whether that pi plus
61:23
equals square root of two was
61:25
greater or or less than uh the rational
61:28
number
61:29
that you that you tell me okay yeah
61:32
uh that it's been a an interesting
61:36
conversation um
61:38
can i just raise my last it's just a
61:40
question my my last year
61:42
is just um if if we were to take this
61:44
formalism of
61:46
of yours i mostly i'd want to know more
61:48
about what it means
61:50
what you require of uh of of math
61:54
whether things have to be finite or
61:56
computable what what does that mean
61:57
does there have to be a polynomial the
61:59
main thing to me is that it's right
62:01
downable
62:02
i'm i'm a right downist i want i want it
62:05
whatever you're talking about write it
62:06
down
62:07
and then and then we can talk about it i
62:10
don't i don't want it just to be in in
62:12
the in the realm of
62:13
of of abstract discourse i want it to be
62:16
to be
62:17
write downable in such a way that we can
62:19
all look at it and and and
62:20
that it doesn't depend on you know our
62:23
way our
62:24
use of hands like this or or or mental
62:26
images or anything like that
62:27
i mean that that's a nice general
62:29
guideline i i just think it's very hard
62:31
you as i'm sure you know like um you
62:34
know you can show that there's
62:35
if you have an algorithm for for
62:38
something you can say oh well you can
62:39
solve this problem but
62:41
but it's very hard to say whether there
62:43
is or isn't you know
62:44
to prove that there isn't uh an
62:46
algorithm to do something
62:47
it's it's it's not so clear to me what
62:50
when you say write down a bull
62:52
you know that leads to a lot of a lot of
62:53
problems i i think you've mentioned
62:55
something about the
62:56
integer complexity you know how much can
62:58
we handle
62:59
what is what is the boundary of this
63:01
theory that you're talking about leads
63:03
to
63:03
uh all kinds of questions that that i
63:06
don't think have been answered
63:08
but but happily we have these brilliant
63:10
computer scientists
63:11
who have been working really hard on
63:14
creating all kinds of beautiful
63:15
mathematical frameworks and i think
63:17
we can anchor a lot of what what we're
63:20
talking about
63:21
by by going to these these these
63:23
mathematical uh
63:25
computer algebra packages and and making
63:28
contact with them and i think that's
63:30
that's ultimately like a really solid
63:32
way of proceeding is is to
63:34
is to pay more attention to what the
63:36
computer scientists are doing and what
63:38
they've been able to accomplish
63:39
and to have a mathematics which is is
63:41
computer
63:42
computationally leaders are real i do
63:46
think that will be a much bigger
63:47
part of mathematics going forward for
63:50
sure
63:51
i think they'd be foolish not to listen
63:53
to that analysts will give them
63:55
some things that are important too i i
63:58
wonder if we've made
64:00
any progress in convincing the other
64:03
of our viewpoints here but at least this
64:05
is a good opening uh
64:07
so you know you i know i think i know
64:09
where you stand maybe hopefully you know
64:10
where i stand now
64:12
yeah what i think are the the issues
64:14
well it's been fun uh
64:16
maybe we can chat again about about some
64:18
other interesting topic at some point
64:20
i'd love to yeah we actually have a lot
64:22
to talk about
64:23
about mathematics besides just this uh
64:25
this issue
64:26
besides this particular issue absolutely
64:29
great thank you thank you so much for uh
64:31
for uh the the pleasant discussion yeah
64:34
great
64:34
okay