Finite versus infinite and number systems
A social scientist wanting to understand the culture of mathematics needs also to have at least some familiarity with the role of number systems throughout history. This is a rich and surprising topics, and in subsequence videos in the series we will be investigating quite different historical roles, starting with the fascinating Egyptian and Mesopotamian number systems and then move forward towards to modern European epochs and eventually the current difficulties that beset the so-called theory of "arithmetic with real numbers".
There are many important sociological questions that arise when looking at the almost deliberate attempt by the academy to ignore discussion of this crucial topic.
The rift between the realities of computational mathematics and the dreaming of pure mathematics in the direction of "real numbers", "infinite series" and "transcendental functions" (just to get started) is now so pronounced that it is become impossible to dismiss. One place to anchor the discussion in the difference between studying a finite discrete system, say a list or sequence , and the corresponding "infinite or unbounded" concepts --which are at their core quite different mathematical objects and need to be treating quite differently.
So let's start a wide ranging discussion about the role of numbers in mathematics, and how to explain the sociological forces that maintain the pretense that all is well despite obvious evidence to the contrary.
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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 (some available only to Members!)
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...
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hello everyone i'm Norman Wildberger and thanks for joining me
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we are talking about the sociology of pure mathematics
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and today i want to start moving towards understanding number systems from an elementary
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social science point of view so while mathematics has its origins in antiquity with geometrical
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interests and astronomical interests in modern times starting with Descartes and Fermat and this
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cartesian revolution in geometry we've moved more and more to founding our subject on underlying
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number systems so we're talking about things like natural numbers integers include negative numbers
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and then also fractions or we say rational numbers but increasingly modern analysis like calculus has
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wanted to invoke other kinds of numbers which are more akin to decimal numbers
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okay but not finite decimals but rather infinite decimals and these are the so-called real numbers
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which now i think it's fair to say sort of form the bedrock of a large part of modern mathematics
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and which i think is so problematic so i think there's a serious discrepancy between what our
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computers can do and what we are claiming that our number systems can do and i want to to show you
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explicitly uh and concretely you know what the the essential difficulties are without getting
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embroiled too much in in complexities of danican cuts and koshy sequences of rationals which i talk
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a lot about in my math foundation series but i want to adopt a more uh elementary point of
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view here and so that you can understand even if you're a non-mathematical type what are the issues
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and in today's lecture i just want to sort of initiate this discussion to give you a framework
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for understanding this very important topic okay so 100 years ago there was a lot of discussion
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in mathematics about foundational issues arising from cantor's theory of infinite sets
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which largely was supported because it was necessary for dedicans theory of these real
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numbers and you should understand that a lot of the support that the modern abstract theory of
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sets and infinite sets gets is because of these foundational underpinnings that it appears to
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give to modern analysis i.e calculus so however as time went by during the 20th century the debate
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subsided and a kind of a an unquestioning uniformity has settled across the subject which is
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very interesting from a sociological point of view okay and really uh ought to be to be studied so
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these days students almost uniformly just accept whatever they're told about about number systems
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and they are you know told well we work with the real numbers even though it's kind of obvious that
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they never actually see an explicit careful laying out of these of these these these real
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numbers and this magical arithmetic that they they ostensibly support and correspondingly a lot of
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students these days are are blissfully unaware of foundational difficulties having to do with
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uncomputable numbers or non-measurable functions or or curious applications of the axiom of choice
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and various anomalies and paradoxes that result from from these things
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so i want to address some of these things and a good place to start i think is with a
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fundamental division between the the discrete and the continuum okay so discrete mathematics
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studies finite patterns and collections and it utilizes natural numbers zero one two three et
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cetera and then negative numbers minus one minus two minus three and ultimately uh fractions also
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uh things that form a over b and the fractions sort of provide a bridge between the discrete
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because they're built up from from natural numbers or integers which have a very discrete aspect
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but they model in some fashion the continuum whatever the continuum actually is so this
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this notion of continuum is hugely problematic as i discuss at length in my famous math problem
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19 series okay i talk a lot about the the core issue with the continuum such a huge problem
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in mathematics so historically the continuum has been accessed in an approximate fashion using
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actually not fractions but rather the the hindu arabic decimal number system where you specify a
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certain number of decimals and the more decimals you have the more accurate your statement is
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and um and that's sort of what your calculator uses and typically your computer these days too
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although there's necessarily an aspect of approximation or round off involved with that kind
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of arithmetic so it's not an exact arithmetic like us pure mathematicians tend to like to think about
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and that's one reason why maybe it it hasn't got an official position
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in the world of pure mathematics even though this applied approximate decimal arithmetic is what
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engineers and scientists use to to create the world and to do all of our fabulous technology
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so pure mathematicians rather rely on a a broader theory of essentially infinite decimals okay
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that's essentially what it is but the infinite decimals have some sort of obvious problems with
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their arithmetic which i'm going to explain to you at some point but pure mathematicians have then
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steered away from that particular point of view towards these things and adopted more
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abstract sort of hard to understand concepts and also harder to use concepts like delicate cuts
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and kosher sequences or equivalence classes of koshi sequences
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of rationals and other things so each of these views has different difficulties but ultimately
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the same essential core problems arise whether you're trying to set up a theory of real numbers
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as infinite decimals or as dedicated cuts or as or as koshy sequences and a lot of the uh
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initial difficulty really comes down to a fundamental problem of how do we specify
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mathematical objects and what the distinction is between a definition and a specification
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so let me illustrate that with some examples so let's say you have a finite sequence let's say
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one five nine one five nine okay that's a little sequence that can be described actually in
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lots of different ways for example uh that's the increasing sequence of possible last digits of
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an integer square if you square a number and it's odd then it will always end in a one a five or a
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nine so that's a way of describing that list or um maybe you could say it's the sequence of numbers
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which are less than 10 which are congruent to one modulo four in other words if you divide by four
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you get a remainder of one that's another way of describing that list and there's there's you know
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many many possible ways we could describe it but a computer scientist realizes that none of those
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descriptions or real or perhaps even definitions really is the essential core the essential core of
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the object is that we want to specify by actually listing the elements okay we list the elements 1
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comma 5 comma 9. that's how we specify really honestly and completely this finite sequence
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now when we make the big jump or attempt to make the big jump to so-called infinite
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sequences okay and i say so-called i put him in quotes like this that means that i don't really
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believe that this is a truly legitimate notion but so i'm playing along so let's say you have a
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another sequence let's say a sequence three five seven and so on so we write three five
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seven dot dot dot okay so then we say okay consider this sequence this infinite sequence
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okay clearly the situation is now dramatically different because we've only specified actually
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three elements and there's essentially an infinite number and
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it's not clear to us what the pattern is we may guess that oh the pattern is we're we're talking
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about odd numbers we're just listing all the odd numbers but maybe we're listing all the odd primes
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or maybe we're listing you know there's all kinds of potential sequences that one could look at that
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start three five seven so in this kind of situation it's not possible for us to
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to list all the elements in such a sequence so we have to adopt a more indirect strategy
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okay and a good way of doing that is to think about a rule or an algorithm or a procedure that
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generates the sequence and define the sequence in terms of that rule or algorithm or procedure
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so that's the way a computer scientist or a programmer would think and that has a lot to to go
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for it but however it adds a lot of complication because now there might be a lot of different
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rules algorithm procedures that specify the same the same sequence and in fact we then
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have to start talking about the difficulties in in deciding whether one particular rule is
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first of all well defined you know because the rule may well involve English statements and
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or or mathematics that's uh that we that's sophisticated that we don't understand
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and then we have to also worry about whether this particular rule and that particular algorithm
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really generate the same sequence or not you see that there's a a world of difference i
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want to stress to you between a mathematics which restricts itself to finite sequences
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and a mathematics which attempts to deal with infinite or unbounded or ongoing sequences
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there's a further complication in that while most computer scientists and scientists would
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would be looking for a rule something that you can write down some finite
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you know set of statements that generates uh such a sequence pure mathematicians for their
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own reasons often want to be more general and or more free they don't want to be constrained by
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a rule this is for various reasons perhaps because they're a little bit nervous about
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having to venture into the land of computer science and and dealing with
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you know the difficulties in defining exactly what an algorithm is and especially how how you decide
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whether two algorithms are the same or not that's hugely problematic but it's also because they
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want to allow themselves the freedom of talking about infinite sequences which are not given by
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such rules that's what this axiom of choice that you may have heard about really ultimately sort
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of comes down to and um so from a computer scientist point of view this is almost like
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some kind of fictitious thing but it's something that that some peer mathematicians um really um
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take as very as being very important so here's a very interesting sociological point of view that
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really should be uh discussed why is there not a a major debate or at least a a a division in
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in mathematics between those mathematicians who want to insist on doing everything in a
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computer science kind of fashion in other words in other words always connecting ourselves with
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algorithms and procedures that can be explicitly written down and run and those pure mathematicians
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who want to not be so constrained okay there should be a debate there should be sides people
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should be on this side of the fence or or that side of the fence and not just some sort of happy
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pretending that we're all in the same boat when we actually believe actually quite different things
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so this all um is directly uh leading to a discussion of of real numbers as infinite decimals
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which is really historically the the core um the core issue here the question is how does one
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attempt to create an arithmetic with infinite decimals can we actually do that or not well
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it turns out that we can't do that but the modern pure mathematician is very reluctant
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to to admit that and to even have a debate about that because it's a kind of a core belief that
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we have to have such an arithmetic because people feel that it's necessary so in the next few videos
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i'm going to show you exactly what's wrong with trying to create an arithmetic eval arithmetic for
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real numbers as infinite decimals and i'm also going to try to convince you that these alternate
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ways like delicate cuts and cauchy sequences and so on really don't solve the problem because
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if you really did have an arithmetic with delicate cuts or with koshi sequences you could transfer it
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to a viable arithmetic with infinite decimals and if sort of that doesn't work then those
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other ones are not going to work either so this is very destabilizing i know modern analysts are
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very low to have this discussion okay and i think it's a really interesting sociological point
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as to first of all why they are loathe and how the academy accepts that such an important issue is
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is deliberately not discussed and the the real truth of the matter is kept from
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from students even this is a crucially important issue i'm Norman Wildberger thanks for listening.