Hi, my name is Bruce Robertson and this is Pirate Philosophy. In this series of videos, I will be describing an original philosophy, one that you won't find anywhere else but it is one that is logical, rigorous and dynamic. Welcome.
So far in this series of videos, we have looked at how, starting from a simple logical processor and sense-data and evolution, an organism or brain can build up a model of the world using the logical process of pattern identification. And in doing this, it creates a pyramid of patterns by a recursive process of making patterns of patterns. And you end up with a model of the world and consciousness and the ability to interact with the world. And so now we come up to perhaps a few 1000 years ago in terms of our evolution to the very start of mathematics.
So what is required or wanted from a philosophy of mathematics? Well, a number of things:
What is mathematics? How is it used? And how does it fit in with the rest of our knowledge of the world and the rest of philosophy?
So having got to the state of evolution of a few 1000 years ago, the human brain is very large, it has got a lot of capacity, ut can do a lot of processing and it's even got spare processing time. It also has a pyramid of patterns and a model of the world and it interacts with the world; but what else can it do?
Well, another thing it can do is to set up what I'm calling an 'abstract system'. And I'm calling it an abstract system to differentiate it from a 'real system'. A real system has a pattern identification process and a model of the world based on sense-data. Then you can have an abstract system, which is entirely abstract; it is separated from a real system and has its own logical processes.
A 'real' logical system is one that has direct links to sense-data.
An 'abstract' logical system is one that is independent of sense-data.
And the basic requirements for an abstract logical system are three things: you need symbols and you need some rules for the manipulation of those symbols and you need an axiom or starting point. From this theorems or inferences can be made.
Basic Requirements for an abstract logical system:
1. Symbols
2. Rules for the manipulation of those symbols
3. Axiom or starting points.
These can then be used to generate theorems or inferences.
One can set up one's own abstract system. I've done so here, where all you have is some symbols, rules and an axiom, For this abstract system that I've created purely for the purposes of this video, it's got four symbols: @, #, $ and %. If you want you can call the abstract system '@#$%'. The rules of the system are very simple rule number one: the string '%@' can be replaced by '@'. Two: strings of the form '%@' or '%#' may be added to each side any string that has $ symbol in it. And third: the string '@@@@@' may be replaced by the string '#' and vice versa. Note that 'string' in this instance refers to a sequence of one or more symbols.
Abstract system @#$%
4 symbols: @ # $ %
Rules of the system:
1. The string '%@' can be replaced by '@'
2. Strings of the form '%@' or '%#' may
be added to each side of any string that has a $
symbol in it
3. The string '@@@@@' may be replaced by '#',
and vice versa
NB 'String' refers to a sequence of one or more symbols.
So, now we want to create theorems from this system and by 'theorem' I mean something that can be deduced logically from the symbols and the rules and the axioms. So, for this system, we can start with an axiom of '@$@' and then we can generate theorems.
Axiom: @$@
Theorems:
@%@$@%@ (by rule 2)
@%@$@@ (by rule 1)
@%@%@$@@%@ (rule 2)
@%@%@$@@@ (rule 1)
@%@%@%@$@@@%@ (rule 2)
@%@%@%@$@@@@ (rule 1 )
@%@%@%@%@$@@@@%@ (rule 2)
@%@%@%@%@$@@@@@ (rule 1)
@%@%@%@%@$# (rule 3)
'@%@$@%@'by rule 2 adding '@%' to each side of the dollar symbol. Then we can condense the '%@' on the right hand side to a simple '@' according to Rule 1. And so on, I won't go through it all but you can end up with the theorem (and these are all theorems) '@%@%@%@%@$#'.
So does this theorem: '@%@%@%@%@$#' mean anything? Is it pointless? What is the purpose? What what makes for a good abstract system? Well, mainly, it needs to be a system that 1. Can generate theorems and 2. It can generate theorems that are 'interesting'. What does 'interesting' mean? Perhaps it means that we can correlate some of the symbols with the patterns from the real world, the patterns within our pyramid of patterns that we have in our own heads.
And we will be coming back to these theorems, but first I want to take a different tack and look at some simple sense-data. So this is for processing in the real world using the normal pattern identification process.
So you're presented with the following data, strings
AFG BRT
LMQ PPR
AAN TOJ
and then I ask you which of the following groups or sets fits best, with the ones above?
1/ FF
2/ B
3/ AZWY
4/ FRZT
5/ NPZ
And I give you five different options. I'm sure you'll have no difficulty in identifying the last one NPZ as being the one that fits with that other group. And why? Well, because you will have identified a concept, the idea of 'threeness', each of them has three different symbols in it. And you've created the idea of three symbols and so the one of the options that has three symbols, is the one that fits with the others. And this concept of 'three' is a pattern; I haven't said anything about numbers, we just have a concept of 'three' and this 'three' is quite often associated with a number '3' in the logical system of mathematics. But it is a slightly different three, because one is purely a symbol within the system of mathematics and the other is a label for a pattern for the concept of threeness. And I would distinguish between them by writing out the one as 'three' and the other one as a simple '3'.
'Three' is a label for a pattern
'3' is a symbol within the system of mathematics
And going back to the abstract system we had of the system #@$%. These are just purely random symbols. Which means that I can simply change them if I want to. So instead of writing '@', I can write a '1', instead of '#' a 'V' instead of a '$' a '=' and instead of '%' a '+'. Then when you substitute those in the system becomes:
Abstract system 1V=+
4 symbols: 1 V = +
Rules of the system:
1. The string '+1' can be replaced by '1'
2. Strings of the form '+1' or '+V' may be added to each side of any string that has a = symbol in it
3. The string '11111' may be replaced by 'V', and vice versa
NB 'String' refers to a sequence of one or more symbols.
Axiom: 1=1
Theorems:
1+1=1+1 (by rule 2)
1+1=11 (by rule 1)
1+1+1=11+1 (rule 2)
1+1+1=111 (rule 1)
1+1+1+1=111+1 (rule 2)
1+1+1+1=1111 (rule 1 )
1+1+1+1+1=1111+1 (rule 2)
1+1+1+1+1=11111 (rule 1)
1+1+1+1+1=V (rule 3)
we get the theorem, one plus one plus one plus one plus one equals V and V of course, is the Latin numeral for five. So effectively, what we've got is an abstract system, which adds five ones together and gets a symbol V or five. And the thing is, with changing those symbols is that those symbols that are '1' and '+' and '=' and 'V' have meaning within the pyramid of patterns, we can map from one system to the other. So that then then they're not isolated solely within the logical abstract system. The relationship between the two can be best described as a mapping and there is a mapping between the symbols of the abstract system and the labels for patterns. So '1', as a symbol, we can associate with a concept of 'oneness'. The '=' symbol we can associate with the concept of 'equals' and so on.
Examples of other abstract systems would be things like Conway's Game of Life,
which has symbols, i.e black and white squares; it has got rules and it's got an axiom i.e. the starting point. And then it goes through a sequence of different applications of the rules to end up in different configurations, or theorems.
So Conway's Game of Life can be considered to be an abstract system, it's a little bit different to the others, because the rules are very specific and they are applied exactly the same way each time it completes a cycle. Whereas in maths or in the other abstract systems I have discussed earlier, there are different rules and you can choose which rule to apply, depending on what you want to achieve.
Another type of abstract system is a game of chess.
It's got symbols, i.e. all the different pieces; it's got rules which are very specific and it's got an axiom i.e. the starting position for all the different symbols. It is different to the other ones, because you have two players, each trying to use the rules in a specific way to achieve a particular end. And each of the different players has a different goal in mind and they are competing against each other and it is a game. But it is a sort of abstract system.
And of course, maths itself is also an abstract system. It has got symbols, rules, theorem and axioms. And going back to the very beginning of maths, it would have started with just mere scratchings on a cave wall or a piece of wood or whatever, with a series of 1111, just a line scratch; essentially, base 1 arithmetic. The example I gave at the beginning, where you have 1+1+1+1+1=V, that is base 1 arithmetic, where each 1 represents 1 and nothing more. I'm using base 1 arithmetic here, because it's the most obvious, it's the simplest and it's the basis for it all. And this is philosophy, I'm not doing maths as such; I'm just looking at the foundations of mathematics and what it is and how it works.
So I'll just do a few examples using base one arithmetic. And, of course, within base one arithmetic, it is very easy to see the correspondence between the symbols and the reality; you have maybe three or four '1's scratched on a piece of wood and you maybe have three or four sheep in a field. And in this way can make sure that you haven't lost a sheep; but you don't need to count them, all you need to do is to establish a correspondence between each symbol of '1' and each sheep in the field.
And I want to show how pure mathematics is the manipulation of symbols, according to the rules to produce theorems and nothing more. There is no meaning to it, it's just meaningless symbols that are manipulated to produce theorems, which may be interesting because they can be mapped onto the real world.
And just to show you how the symbols can be manipulated, I will go over very quickly a few examples using base one arithmetic; where you just have strings of '1's or an empty string.
An algorithm is a set of rules for generating a theorem
So if you want to add two base 1 numbers together or base 1 strings, there is a simple algorithm that you can use. I will put one here:
Addition: To add Strings S1 and S2 together and put the sum in S3
Set S3 = ' '
Add S1 to RHS of S3
Add S2 to RHS of S3
Output S3
S1, S2 and S3 are simply labels for strings. That's all they are.
So you start off by setting S3 equal to the empty set or nothing. Then you add S1 to the right hand side of S3 and then S2 to the right hand side of S3 and then you output S3 as your theorem. You can see how this works in a worked example:
Worked example: Add 111 to 11
S1 S2 S3
111 11
111 11 111
111 11 11111
Output 11111
So you want to 'add' the two strings '111' and '11' so you set S1 to '111', S2 to '11' and S3 to ' '. Run them through the algorithm and the output is the string '11111'.
And this output theorem is interesting, because it matches what we know about the real world, where the concept of 'two' combines with the concept of 'three' to give the concept of 'five'.
And then you can use another algorithm for 'subtraction'. If you want to subtract S2 from S1, here is an algorithm for that:
Subtraction: Subtract S2 from S1 and return a '-' sign with the string if S2 is greater than S1
Set Ss = '-'
A If S2 = ' ' Goto C
If S1 = ' ' Goto B
Remove last (RHS) symbol of S2
Remove last (RHS) symbol of S1
Goto A
B If S2 ≠ ' ' Set Ss to '-'
Set S1 equal to S2
C Output: Ss S1
Note that this one returns a negative sign if the length of the string S2 is longer than the string S1. so if you were wanting to subtract four from three, it will return -1.
So here's a little worked example. Subtracting '11' from '1111': output '11':
Worked example: Subtract 11 from 1111
S1 S2 Ss
1111 11
111 1
11 1
Output 11
And if you do the reverse and subtract '1111' from '11', if outputs '-11'.
Second worked example: Subtract 1111 from 11
S1 S2 Ss
11 1111
1 111
11
11 11 -
Output : - 11
And that negative symbol is just a symbol, it means nothing within the abstract system of pure mathematics, but in the real world, we can associate that symbol as being 'negative'.
And then there is an algorithm for the multiplication of two numbers or of two strings.:
Multiplication: Multiply S1 by S2 and puts the product in S3
Set S3 = '-'
A Add S1 to RHS of S3
Remove last symbol of S2
If S2 ≠ ' ' then goto A
Output S3
And here is a worked example:
Worked Example: Multiply 111 by 11
S1 S2 S3
111 11
111 1 111
111111
Output 111111
And note, that when I'm doing all this, there is no use of number within the algorithms, I am merely manipulating the strings; nothing more; i.e. adding a symbol here, taking one off there, that's all. There is no concept of number. It's just a manipulation of strings.
And here is an algorithm for division. Division can be associated with repeated subtraction and so this one uses the subtraction algorithm used previously:
Division : Divide S1 by S2 and return a fractional part if the numbers do not divide exactly
NB This algorithm uses the subtraction algorithm (described above) as a subroutine.
Set S3 = ' '
Set Ss = ' '
Set S4 equal to S2
If S2 = ' ' Output ERROR then Stop
A Subtract S2 from S1 (Using the subtraction algorithm that returns Ss and S1)
Set S2 equal to S4 (this just resets the value of S2)
If Ss = '-' Goto B
Add a '1' to the RHS of S3
If S1 ≠ ' ' goto A
Output S3
Stop
B Subtract S1 from S2 (using the subtraction algorithm)
Output S3 + S1/S2
Stop
It shows how S1 can be 'divided' by S2. using nothing more than string manipulation. It is reasonably comprehensive in that it returns a fractional part if the numbers do not divide exactly. It also returns an output of 'ERROR' if S2, the 'divisor', is empty.
I've got some examples here of how the different strings change as you go through the algorithm.
Example Divide 111111 by 11
S1 S2 S3 S4 Ss
111111 11 11
1111 11 1 11
11 11 11 11
11 111 11
Output 111
Second worked example: Divide 11111 by 11
S1 S2 S3 S4 Ss
11111 11 11
111 11 1 11
1 11 11 11
1 11 11 11 -
Output 11 + 1/11
Third worked example: Divide 11111111 by 111
S1 S2 S3 S4 Ss
11111111 111 111
11111 111 1 111
11 111 11 111
1 111 11 111 -
1 11 11 111 -
Output: 11+11/111
So if one 'divides' '111111' by '11', the output ls the string '111'. And if one 'divides' '11111' by '11', the output is '11 +1/11'. And if one 'divides' the string '11111111' by the string '111', the algorithm outputs the string '11+11/111'.
And this demonstrates how base 1 arithmetic can be used to denote fractions (and by extrapolation all rational numbers).
And you might ask: What does base one arithmetic have to do with ordinary mathematics, where we deal generally in base 10? Well, it is quite simple, again using only the manipulation of strings, to convert base 1 strings to base 2 numbers.
And I have here a little algorithm that will do that. It converts base 1 numbers to base 2:
Algorithm for converting Base 1 to Base 2 arithmetic. Using only the manipulation of strings.
S0,S1,S2.S3 are stores for strings of symbols.
Let S0 = the base 1 string to be converted
Set S1= ' ', S2 = ' ', S3 = ' ' (' ' denotes blank or empty)
Set S1equal to S0
A If S1 = ' ' then add a '0' to LHS of S2 then goto B (LHS means Left Hand Side)
If S1 = 1 then add a '1' to LHS of S2 then goto B
Else Remove last (RHS) symbol of S1
Remove last (RHS) symbol of S1 (this is a repeated operation)
Add a '1' to LHS of S3
Goto A
B If S3 = ' ' then goto C
Else Set S1 = S3
Set S3 = ' '
Goto A
C Output: S0 base 1 converts to S2 base 2
Stop
And here is a little worked example of how the number 11111 base 1 converts to 101 base two.
Worked example: Convert 11111 base 1 to base 2
S0 S1 S2 S3
11111
11111 11111
11111 111 1
11111 1 11
11111 1 1 11
11111 11 1
11111 1 1
11111 01 1
11111 1 01
11111 1 101
Output: 11111 base 1 converts to 101 base 2
So they're both five, that is '5' in base 10. So that's five base one converts to five and base 2. And of course, if the base 1 number, you are wanting to convert to base 2 contains fractions, you can just convert the numerator and the denominator separately.
Then having got to base 2, you can, of course, convert to base 10; there are a lot of algorithms around that will do that.
I've gone over some of this in some detail, because there's very little on the internet, or anywhere about how base 1 arithmetic works. So I've had to go into that in a little bit of detail.
But what we have been over is the start of mathematics. And it's the start of how to understand mathematics as string manipulation of abstract symbols.
And this is also evidenced by the fact that pocket calculators and computers can do arithmetic. They use simple algorithms, manipulating strings; the strings mean nothing to them, other than merely the significance of the symbols within the computers, and how they are specified within the rules of how the strings are manipulated. Otherwise, it it completely meaningless to a computer or a pocket calculator.
It is only when that output is taken and put across to a person's pyramid of patterns that we ascribe meaning to it, and we associate the output with our concept of 'numbers', the concept of 'equals' and the concept of everything else.
And it is interesting to note that intrinsically, within an abstract system, there is no requirement for what is called consistency, because they are just meaningless symbols. The only requirement or expectation of consistency comes when you look at how you might map it to the real world and then you want to say well, does it make sense within the real world? So for example, if the string came along of '1=2' (which incidentally is quite easy to do, using simple algebra):
'Proof' that 1 = 2
Let a =b
Then a²=ab (multiplying each side by a)
then a²-b²=ab-b² (subtracting b² from each side)
then (a+b)(a-b) = b(a-b) (factorising each side)
then a+b = b (dividing each side by (a-b))
then b+b = b (substituting a for b since a = b)
then 2b = b (combining terms)
then 2 = 1 (dividing each side by b)
or 1 = 2
It doesn't map well to the concepts that we have of the world, given the typical mapping between numbers and our concepts. Our concept of 'one' is quite different to the concept of 'two' and we don't like to think that 'one' equals 'two' and so we don't like it when our mathematics produces the string '1=2'.
So mathematicians put in a rule that you're not allowed to divide by zero. And while this rule may seem to be somewhat arbitrary and contrived, it does succeed in invalidating the 'proof' of 1 = 2. (In line 4 of the 'proof' above there is a division by a-b when a=b and thus a-b is zero and thus this line is invalid).
The thing about abstract systems is that they are totally logical and totally rigorous, within the rules of the system. It's not like the real world where you' have sense-data coming in and it's not quite clear what that sense-data is and you're not quite sure what is the best pattern that fits the data.
In an abstract system, it is totally rigorous. As previously mentioned, you have symbols, rules, axioms, theorems and those are totally rigid. A lot of people think that mathematics is 'true', because within that system, you can prove the theorems in the way that we just proved that '1+1+1+1+1 = 5', you can also prove that '1+1 = 2' and it's all totally rigorous within that system. and some people like to say that means it is 'true'; but if it is 'true' then it' is only true within that system. It's only true within that particular use of symbols within those particular rules and so it's totally rigorous and 'true' but only within that particular abstract system.
And that is the case with the mathematics abstract system too. What is logically proven within that system is logically 'true' but only within the system of mathematics.
So, now, we come to the four distinct components of mathematics which is the title of this video: Pure maths, Applied maths, Design maths and Pattern maths.
1. Pure maths is the pure abstract system.
Pure Maths: The manipulation of symbols according to the rules of the system to generate theorems.
It is the what we talked about before and which is exclusively concerned with the symbols the rules, the axioms and the theorems. And it is all totally meaningless. It is only concerned with the manipulation of strings of symbols. They are just strings, they mean nothing else. There is no question about consistency or anything like that; whatever the rules dictate that is what generates the theorems within the particular system of pure mathematics. What is variable in the course of the system of mathematics is how those rules are applied, in which order, and with what axioms you want to apply. So, for example, if you have simultaneous equations as your input you can choose which rules you are going to use to manipulate those equations to get you to where you want, i.e. to a 'solution' of the equations.
2. The second one is Applied Maths, which is the application of pure maths to the real world.
Applied maths: The application of the theorems of pure maths to the real world by means of a mapping.
Suppose we have a field and we want to know what the area of the field is. What we can do is to map between measured dimensions of the field in the real world and particular symbols or numbers in the abstract domain and then use the manipulation of those numbers in the abstract world to arrive at a theorem which we can then map back to the real world.
But the mapping between the abstract world and the real world is not always obvious. So for example, if you're counting sheep, you've got 1+1=2 and you can map that to 1 sheep + 1 sheep = 2 sheep. But if have drops of water, like one drop of water plus one drop of water generally doesn't produce two drops of water; it just produces one slightly larger drop of water. If you wanted to model the motion of a ball traveling through the air, then you don't want to use statistics or matrices, what you want to use is calculus and so on.
(N.B When I refer to the 'real world', I am more specifically referring to the model of the world that we have in our minds and which was created through a process of pattern-identification, as discussed in previous videos.)
Mathematics is incredibly complex. While we can learn the game of Conway's Game of Life or chess in a few minutes; it takes 10 or more years in school to learn mathematics. Even then, one only learn a small portion of mathematics. It is very, very complex; but extremely useful, particularly in its mapping of statistics, physics, finance and all those sorts of things. Mathematics is very, very useful but very complex.
3. Number three: Design maths.
Design Maths: The design of the abstract system, including the creation of the rules and symbols.
This is the where the rules for the mathematical system are chosen. ie how the symbols are manipulated within the abstract system. It is where the rules are selected or designed, for example, for solving different equations and within calculus, simple algebra and geometry. What are the rules? How can the rules be designed so that the theorems can be mapped in a meaningful way to the real world? In the real world we have concepts of two dimensional space, three dimensional space, money, large groups of people, the physics of the real world and so on. So there is a lot in the design of the pure mathematical abstract system to arrange it so that many of its theorems can be mapped to a model of the real world. Mathematics is specifically designed and has specific rules so that it can be used to model the real world.
And also within the design of maths, one has to invent new symbols. For calculus uses specific symbols that are different to any other. And there is also a special symbol (i)that is used to designate the square root of minus one. And that is a very useful symbol but it had to be designed. It also had to be designed as to how that symbol and its associated rules can be brought into the system of mathematics without creating a contradiction, something which does not match with the real world; and that is quite interesting. And while the square root of negative one is called 'imaginary', it's really no more imaginary than ordinary integers. You can count sheep using integers..1, 2, 3, 4, 5; or if you wanted to, you could count them using imaginary numbers of 1i, 2i, 31, 41 etc. it makes no difference it is just a different mapping.
4. The fourth component of mathematics is what I call pattern mathematics.
This component of mathematics is really just for mathematicians. Mathematicians look at the whole system of mathematics and look for patterns within the theorems.
Pattern Mathematics: The application of pattern-identification logic to the theorems of maths to search for patterns and ask questions.
One can look at all the theorems of maths and then search for a pattern, using a pattern-identifying process. Then one ask questions such as: 'Are there a finite number of prime numbers'; 'Is there a solution to the equation : A³ + B³= C³, where A, B and C are integers?' and other questions like that. Mathematicians enjoy playing around with questions like these; they explore the whole system of mathematics.
But this is not directly related to the other three components of mathematics.
This approach to the philosophy of maths is entirely consistent with Gödel's Incompleteness Theorem.His theorem entailed making assumptions regarding the completeness of mathematics and that one can make statements about mathematics that are not theorems, and of'truth' of such statements, and then showing that by making these assumptions there was an inconsistency.Which then implied that at least one of the assumptions was incorrect. And the one normally taken to be incorrect is that of completeness; i.e. the system of mathematics is incomplete.And some people consider that to be a problem; but within the approach to mathematics that I have been describing, it is not a problem at all. There's no question about whether the system of mathematics is complete or not; it just is what it is.
Gödel's Incompleteness theorem is entirely compatible with The Pattern Paradigm approach to mathematics.
The only place that truth appears within this approach to mathematics occurs when a theorem is generated and then that theorem can be considered to be 'true' within the system and one can then label it, if one so wants, as 'true'.
So I am claiming that this approach to the philosophy of mathematics within the overall philosophy of The Pattern Paradigm, which I've been describing and will continue to describe, is simple, comprehensive and accurate.
And this is in contrast to the Standard Western philosophy approach, which is, to say the least, problematic. The difficulty that Standard Western philosophy has is that there is no clear distinction between the pure maths and the applied maths; they put the two together; and then they try to justify a certain truth of the whole thing and they try to prove that it's founded on some pure logic or something and it really doesn't work. Bertrand Russell and others tried to prove that 1+ 1 = 2 was a consequence of pure logic; and, of course, it didn't work. And now other people try to claim that mathematics can be deduced from sets, thinking that sets are somehow more fundamental than integers. All they have really shown is that there is a correspondence between sets on one hand and pure numbers on the other; they haven't shown that one is more fundamental than the other.
As a final note, might, some of you might find it interesting that the thumbnail and the picture at the beginning of this video was created as a variation of a Mandelbrot set and I think it's an interesting picture. It's a way that shows the complexity of mathematics on a very pure level to produce something that is interesting.
Well, that is all I have for you today. If you have any interesting comments or questions about today's video, please leave them in the comment section below and if you would like to continue this journey with me, then please subscribe to my channel, give it a thumbs up and ring the bell.
Thank you
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