Is pure mathematics logically viable? Five Challenges!

Some tough talk directed towards the professoriat and students of the subject: is it time to re-evaluate what exactly is going on in Pure Mathematics?

This is part of a series on the Sociology of Pure Mathematics, where we try to delve into and unravel some of the mysteries of the profession from a social point of view.

https://www.youtube.com/playlist?list...

Video Content:

00:00 Introduction

1:31 Challenge 1

4:54 Challenge 2

8:08 Challenge 3

13:44 Challenge 4

17:00 Challenge 5

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

Year 9 Maths: https://www.youtube.com/playlist?list...

Ancient Mathematics: https://www.youtube.com/playlist?list...

Wild West Banking: https://www.youtube.com/playlist?list...

Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...

Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?

list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov

Math History: https://www.youtube.com/playlist?list...

Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...

MathFoundations: https://www.youtube.com/playlist?list...

Wild Linear Algebra: https://www.youtube.com/playlist?list...

Famous Math Problems: https://www.youtube.com/playlist?list...

Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...

Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...

Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

Differential Geometry: https://www.youtube.com/playlist?list...

Algebraic Topology: https://www.youtube.com/playlist?list...

Math Seminars: https://www.youtube.com/playlist?list...

And here are some of the other Wild Egg Maths Playlists:

Solving Polynomial Equations: https://www.youtube.com/playlist?list...

Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

Algebraic Calculus One: https://www.youtube.com/playlist?list...

Algebraic Calculus Two: https://www.youtube.com/playlist?list...

0:11

hello everyone i'm Norman Wildberger today we're going to be continuing with our sociology and

0:15

pure mathematics series where i'm trying to have a larger look at the sociology of pure

0:23

mathematics what's really going on there in terms of underlying assumptions and actually the logical

0:28

integrity of the subject i think there are really serious issues that the professoriate is working

0:35

hard not to look at so today we're going to be looking at five challenges in the direction of

0:41

whether the subject's actually logically viable so more or less pure mathematics is regarded by most

0:48

people as sort of the apex of logical correctness and clarity and rigor but is this really justified

0:57

in my view pure mathematics is actually almost a belief system it's a belief system unfortunately

1:05

resting on very dubious foundational assumptions and definitions and today i want to illustrate

1:14

that in a very direct way by giving my colleagues five very explicit concrete challenges that shed

1:24

light on the very significant weaknesses currently besetting the subject so our first challenge will

1:31

not be a surprise to the regular listeners to my channel it's a challenge that i've presented

1:36

on many occasions and probably will still have to okay and for some curious reason no one seems to

1:42

be able to adequately meet this challenge hmm okay what is it well it's to simply calculate the sum

1:49

of arguably the three most common irrational numbers which are pi e and square root of two

2:00

so in the usual story analysis is built up from so-called real numbers and they're arithmetic

2:06

these are arguably the simplest or most common real numbers which are not rational numbers

2:13

arithmetic consists of various operations but addition is the simplest one so let's apply the

2:18

simplest operation to the three arguably simplest irrational numbers what do you get when you do

2:25

that what is the answer to the question what is pi plus e plus square root of two well let's all

2:32

agree first of all that if your answer to this question is oh the answer is pi plus e plus

2:39

square root of 2. if that's the answer then we can all admit i think that you don't really have

2:47

a proper theory of arithmetic with real numbers your arithmetic with real numbers is really fake

2:54

in the same way that if you went into a primary school classroom and you asked the kids who have

2:58

been studying rational numbers children what is the sum of one-half plus one-third plus one-fifth

3:05

and if all that you got back was oh it's one-half plus one-third

3:09

plus one-fifth you know that these people don't really understand this arithmetic now

3:19

the fact that you can't get a better answer than this and if you can get a better answer

3:24

then please let us know what it is but let's make sure that we understand that i want an actually

3:30

correct answer like not a roughly correct answer i want an exact correct answer completely clearly

3:37

so we can all look at it and so that we can check it okay and you perhaps are not able to do that

3:44

in which case i think you have to admit that the foundation of analysis which is arithmetic

3:50

with real numbers is dubious it's flawed and therefore consequently all these theories which

3:58

are built up from the arithmetic with real numbers they are also logically compromised

4:05

mathematics is a sort of a relentless mistress in this regard right if you

4:09

don't have the the foundations of a subject correct if there's weakness or ambiguity or

4:18

logical error in the underlying framework for a subject then everything that you built on top

4:24

of that no matter how sophisticated no matter how clever you are is going to be compromised

4:30

so it's not just modern analysis unfortunately also differential geometry is largely built on

4:35

real numbers topology is largely built on real numbers many other fields also

4:40

all these fields necessarily deflate once we admit to ourselves that the arithmetic with

4:46

real numbers is really a kind of a fake arithmetic and that is the reality of it

4:54

challenge number two also seemingly very simple it's just to calculate sine of five

5:03

so the usual story is that we have functions in analysis and the sine function is a particularly

5:11

sort of elementary simple kind of building block for a lot of other things so sine is supposedly a

5:17

function which inputs real numbers and outputs real numbers okay here we're inputting just

5:24

natural number five so it should be particularly easy we want to compute what sine of five is

5:30

and when i say compute i mean exactly precisely completely it's not going to suffice i hope you

5:39

realize just to get your calculator out and to press the sign button and the five button and

5:44

sort of see what the calculator spits out the calculator will spit out 8 or 16 or whatever

5:49

decimals okay but i think we can all agree that that's not really sine of five that's just an

5:56

approximation to sine of five the question is how do you actually calculate sine of five

6:02

and if you can't do this ultimately if you can't do this then you're really not in possession of

6:10

a valid theory of functions all right the core aspect of a function is that you input

6:15

something and you actually get something out and if you're not getting something out or if you're

6:20

only getting it out approximately then it's not a real function it's not a legitimate mathematical

6:26

object of course in applied mathematics that's fine you know we can accept that

6:32

the calculator is giving us something very useful and you know important for practical applications

6:39

but in pure mathematics we want to have exact theories and so this question of whether we

6:44

can compute this exactly is very pertinent now there's nothing special about sine i could have

6:50

used log tan the gamma function the zeta function arc cosine not the gamma function for integer but

6:57

the gamma function for some non-integer value the zeta function are cosine hundreds of other

7:02

functions it's all the same story pretty well okay almost none of these things can in general be

7:09

calculated exactly in terms of you know inputs and outputs and so our whole understanding that

7:17

you know modern mathematics is largely built on functions really is highly questionable okay

7:24

and if you watch the algebraic calculus course especially the algebraic calculus 2 lectures

7:30

and if you've been watching some of my series in the wild egg maths exploratory

7:36

videos where i talk about orthogonal polynomials and power series and poly series you'll see that

7:43

there are legitimate alternatives to this functional fixation that we currently have

7:50

that have been developed by the combinatorialists okay so there's there's an alternative to this

7:55

story and we should really be looking at it very seriously so that's challenge number two please

8:02

tell us what exactly is sine of five our third challenge to exhibit an interesting infinite

8:11

topological space when i say interesting i mean one that's sort of not just sort of trivial so in

8:19

20th century mathematics this subject of topology perhaps initiated in the later part of the 19th

8:25

century by Poincare more or less emerged as quite a an important discipline and at the heart of it

8:33

is this idea of a topological space which tries to embody somehow our intuitive notion of continuity

8:41

if you've watched my famous math problem number 19 you'll know that this problem of continuity is

8:47

really an absolutely fundamental core problem in mathematics trying to model what we really mean by

8:53

the continuum so the 20th century has come up with this notion of a topological space which these

8:59

days is accepted almost universally as sort of you know the de facto you know way of proceeding

9:05

in this direction but actually it's highly problematic and uh so let me illustrate that by

9:12

sort of explaining at least illustrating what a finite topological space is to give you the

9:17

main ingredients okay so a finite topological space consists of a set in this case a set of

9:24

numbers one two three four and five so it's just a finite set five things and then the essence of

9:29

it is a set of so-called open sets so what we're going to be looking at is some subsets of our set

9:37

s okay so here are a bunch of subsets of this set s there's a subsets the empty set is the subset

9:45

just consisting of this element one here's a subset consisting of two and five etcetera

9:50

so this collection of subsets has some important properties that characterize

9:58

what are called open sets okay so what are these properties well the properties are that

10:03

if you take the union of any two of these you're going to get another one in this collection

10:11

so if you take the union of this set and this set you're going to get the set 125 and that should be

10:16

somewhere oh there it is there okay so the union of these two is there take the union of these two

10:22

two five two three four take the union you're gonna get two three four five which should be

10:27

somewhere there it is there okay etc and also this should be closed under intersections

10:34

so if you look at the intersection of any two sets in this list it should also be there so

10:39

for example the intersection of this set and this set the things that are common to both

10:44

just the element two so there it is there that's the intersection of these two if you

10:49

take the intersection of this one and uh and say this one well it's just this one itself if you

10:55

take the intersection of two and one it doesn't have any intersection so that's the empty set

11:00

okay so this is an example of a finite topological space and it's you know not that onerous because

11:10

we have a relatively small number of elements sort of at the base of things now the theory of finite

11:17

topological spaces is not particularly developed people don't pay that much attention to it because

11:24

if the set gets big okay the amount of data required to list open sets in general and to

11:31

check that unions and intersections are always closed that's a lot of work and so this is

11:39

curiously a theory which almost is invisible at the finite level okay but nevertheless it's this

11:47

theory when we jack it up to so-called infinite sets which forms the the basis for our modern

11:53

understanding of continuity so the challenge is to show in an explicit fashion like this

12:02

to show us a single non-trivial example of an infinite topological space now i have to say

12:11

that in that case uh for various reasons there's a little bit of uh sort of extra

12:17

requirements that that are required here so um for an infinite topological space the requirement

12:24

is that we should um we should be able to take unions not just of two things or finite number

12:30

things but we should be able to take unions of infinite number of things so in the usual

12:35

imagination you imagine an infinite set okay and then you have these subsets consisting of the open

12:41

sets and there's typically an infinite number of them okay it's an infinite number of infinite um

12:46

well most of the subsets not all of them but but a lot of them will be in infinite themselves okay so

12:51

you've got this huge class of or set of of subsets of this infinite set that you're starting with

12:57

and then you have to check these various union and intersection conditions but the

13:01

union condition is a little bit you know bigger it's uh requiring that arbitrary unions even if

13:09

you take the union of an infinite number of um open sets you're still going to get an open set

13:16

so to actually exhibit something like this and to check that you you really have a viable system of

13:21

open senses in my view simply something that's far beyond our capabilities right in reality we're not

13:29

able to exhibit such a thing but i'm willing to be proven wrong okay so i'm willing for someone to

13:34

come along and in a real honest fashion to exhibit a single interesting infinite topological space

13:43

our fourth challenge directly challenges a fundamental belief shared by almost all pure

13:50

mathematicians that's in a result called the fundamental dream of algebra no sorry

13:54

the fundamental theorem of algebra okay so the challenge is to factor

14:00

this polynomial x to the seven minus two x plus three into real linear and quadratic factors

14:10

i've pretty well just made this thing up okay and i'm confident that uh you're not gonna be able to

14:15

do this but um but i could be wrong okay but um so that's the challenge according to the fundamental

14:23

theorem of algebra uh any such polynomial factors into linear and quadratic factors despite all the

14:32

evidence to the contrary you know if you actually write down some polynomial you know unless you've

14:38

chosen a very special one typically of low degree you're going to find that it doesn't

14:42

factor into linear and quadratic factors this is a remarkable a remarkable situation that we've

14:49

got this very direct theorem which is sort of obviously wrong okay and that's in that you just

14:56

give us some random example and you're not able to do what the theorem claims that you can do

15:02

and when i say factoring it i mean like an honest factorization okay so that you actually

15:07

have factors a linear factor means something in the form a plus b x where a and b are numbers

15:14

quadratic factor means something of the form a plus b x plus c x squared where a b and c

15:22

are numbers okay so you have to cook up these linear and quadratic factors and then i'm going

15:27

to multiply them all together okay to check that at the end of that you're going to get x to the 7

15:33

minus 2x plus 3. and i claim you're not going to be able to do it in direct in direct opposition

15:42

to this fundamental claim that lies at the heart of a lot of modern pure mathematics

15:49

so it's a pretty interesting sociological question how such a state of affairs can go on and how

15:54

like the entire you know academy just willfully refuses to to even look in this direction

16:01

we just repeat this everybody repeat the fundamental theorem of algebra is true

16:06

fundamental theorem of algebra is true all evidence to the contrary disregarded

16:13

once you accept this or see it well then you realize that a lot of modern algebra

16:19

indeed geometry sort of deflates it doesn't disappear but none of these subjects disappear

16:26

it's just that the results are not really true the results are just in the form of preliminary

16:33

thinking okay they are sort of waiting to be resurrected taken out of the recycling bin

16:40

and analyzed more carefully more precisely and given more solid foundations

16:49

huge amount of work for the next generation of pure mathematicians okay so there's just so much

16:56

to do here once you realize the true state of affairs and so for our fifth and final challenge

17:02

we moved sort of to algebra algebraic geometry which on the face of it might seem to be a you

17:08

know very down to earth field but it's it's not so the challenge is to exhibit even one non-trivial

17:16

infinite variety so variety over the over the real numbers or the complex numbers with not

17:22

just you know a single point in it or something okay so what is a variety well a variety is

17:30

a set of points typically in some linear space or maybe a projective space but let's just stick

17:36

to the linear case so maybe a triple of complex numbers x y z such that the x y z coordinates

17:43

satisfy say an equation like this x squared times y to the seven plus y cubed plus two x to the

17:49

fourth z to the fifth equals seven so you look at the set of all complex numbers of this kind

17:56

that satisfy this equation now according to the usual story this is an infinite set because

18:02

well you can basically just assign random values to any two of them okay like you could assign

18:08

a random value to x into z and then you get a cubic equation and y and according to the

18:13

fundamental theorem this cubic equation will always have have zeros and so you'll be able to

18:19

find triples that extend the the given values that you've plugged in for x and z

18:25

so from that point of view there's always an infinite number of elements in this set

18:30

and and that's then the variety so my my challenge is to exhibit such a thing

18:38

now if you're an algebraic geometry you'll say wait a minute isn't isn't this

18:42

an exhibition of it i mean isn't this the variety and i want to emphasize that no that's not okay

18:49

the the definition of a variety is not as um you know or a string of things that that evoke

18:56

something it's actually the set itself so this is to be thought of as a description of a set

19:04

this is not a variety this is just a description of a variety

19:08

to actually calculate the corresponding variety you actually have to do a lot of work you have

19:13

to actually find these triples that satisfy this equation and unfortunately not just 10

19:19

or 20 of them you're obliged to find all of them okay you're obliged to find all them

19:26

so i want to um just really emphasize the difference here between this description and

19:32

and actually finding the object suppose that your company you're working for a

19:37

census company and you've just been hired by the government to perform a census and to get a list

19:43

of all names of all people who are in new south wales Australia on june 1st of some some year okay

19:49

maybe next year okay so your company's gonna have to do a lot of work uh you have to go out there

19:55

and you're gonna have to find out who all the people are gonna make a list of all the names

20:00

but you are just a recently graduated pure mathematician so so at the meeting of this

20:04

team you stick up your hand and you say hey i've got a good idea why don't we just define

20:09

p to be the set of all names of all the people in new south wales and uh okay so people say

20:17

okay we can do that and then you say okay then we're done right that's what we want we want p

20:23

p is the set of all names of all people in new south wales so there it is that is the set

20:30

well if you tried to do this in real life of course you would be laughed at immediately

20:34

right that's not the set that's a description of the set that's you know an indication of where we

20:40

want to go but there's still a lot of leg work to be done before we actually get that list of names

20:46

and it's very similar here with this variety story this is just a description

20:50

okay the challenge is to find an example it doesn't have to be this one make up

20:55

some other one where you can actually exhibit all the elements of this variety

21:04

and if you can't do that well then algebraic geometry deflates doesn't it

21:10

so i actually could come up with quite a few more such

21:14

such problems and challenges and maybe in some future video i will

21:18

already do that but i would like to really encourage mathematicians who are out there

21:22

to actually meet these challenges you know like step up to the plate grab the bat and have a swing

21:29

tell us how you actually meet these challenges suppose that something important depended on it

21:37

i'm Norman Wildberger thanks for listening.