[00:03] about it that stimulates my brain in just the right way to make all the nice Feelgood chemicals come out it gets me in the same way as solving a puzzle
[00:16] in the same way as solving a puzzle Thrills someone else however I have two massive problems when it comes to mathematics the first is I have a tiny bit of doubt that under all of my love for mathematics is a strange form of
[00:31] performance and I don't really actually love mathematics at all but it's all a massive act but apart from that impostor syndrome I have a more fundamental
[00:43] syndrome I have a more fundamental problem I do not know what a number is now another Ruth has done a wonderful video titled what is a number it is a
[00:55] video titled what is a number it is a perfectly reasonable video doing all that that video wants to do however it doesn't actually answer the question in doesn't actually answer the question in its title the video is a kind of lie and
[01:09] its title the video is a kind of lie and I suspect that Alex of another roof knows that it's a lie though not the type of evil lie that is attempting to type of evil lie that is attempting to mislead but more lies told to students
[01:22] where you simplify a complex subject by leaving out the things that would leaving out the things that would distract or confuse now in in this otherwise totally acceptable video Alex answers the question by showing a way to
[01:37] construct the counting also called the natural numbers from sets in set theory he then goes on to explain how to build up from the counting numbers to the whole numbers from the whole numbers to the rational numbers and on to the badly
[01:54] the rational numbers and on to the badly named reals and the not so poorly named complex numbers to explain why I feel this answer is unsatisfactory I am going to use an analogy imagine if you asked a
[02:10] analogy imagine if you asked a structural engineer what a house was and they answered by telling you how to build a house you understand why that misses the point of the question you can't just point to the result of
[02:24] following a building plan and say that's what a house is I want to know how to distinguish between things that are houses and things that are not houses what is even worse with Alex's explanation for me is that I am cursed
[02:41] with knowing a little bit too much to continue with the house building analogy when the structural engineer explains how to build a brick house since I know that there are houses made of wood and houses made of straw I also think that
[02:57] his definition of a house might be lacking what Alex gave was a wonderful explanation of how to construct the counting numbers from
[03:09] construct the counting numbers from nested sets these are called Von Newman that name kind of gives the game away doesn't it if this was the only way of constructing counting numbers wouldn't we just call them the counting numbers
[03:24] we just call them the counting numbers if we have to give it a name to distinguish it from from other constructions that's a clue that other constructions exist an alternative construction of the counting numbers
[03:40] that I'm aware of is called The Church numerals not because these numbers are religious the religious numbers of course are the cardinal numbers who are responsible for the election of the math
[03:54] responsible for the election of the math Pope the church numerals are named after Pope the church numerals are named after Lon o Church the professor who supervis Lon o Church the professor who supervis Alan turing's PhD and is almost as
[04:08] responsible for the field of computer science as touring is the church science as touring is the church numerals are an alternative system of encoding the counting numbers using Lambda calculus which I'm not going to
[04:23] go into depth about because if you started me talking about Lambda calculus it would be impossible to tell when I would stop but it is a way of would stop but it is a way of representing mathematics and computation
[04:38] in the form of a system of string substitutions but that's not the only system that you can construct to model the counting numbers when I did my video the counting numbers when I did my video on gender and fixed Point logic I
[04:53] briefly mentioned that you could create the counting numbers by taking the fixed the counting numbers by taking the fixed point of the maybe type indeed it is really super easy to create new ways to
[05:06] construct the counting numbers sometimes I've even done this for fun the thing is that I am sure that Alex from another roof knows all of this just as well we
[05:19] all have to make decisions about what we explain and what we leave out otherwise we would end up with videos that were overwhelming and didn't make sense the
[05:32] thing about all these different constructions of the counting numbers is constructions of the counting numbers is that they all follow the same rules they that they all follow the same rules they behave in the same way perhaps rather
[05:45] than defining numbers based on how they are constructed we should Define them on are constructed we should Define them on how they behave this is the solution to how they behave this is the solution to this problem that jepe piano decided on
[05:59] this problem that jepe piano decided on the the following rules are called the the the following rules are called the piano axioms for counting numbers Zer is piano axioms for counting numbers Zer is a counting number we don't Define what
[06:12] a counting number we don't Define what zero refers to it's just zero for my postmodernist friends it is a floating postmodernist friends it is a floating signifier sometime times the piano axom
[06:25] start at 1 which personally is very odd to me but I'm not going to attach somebody else's culture a number system with just zero is kind of empty and not really useful Let's Get Away of introducing new numbers if square is a
[06:41] counting number then s square is a counting number s stands for successor in other words the next number after even though it is clearly intended for
[06:55] even though it is clearly intended for s0 to be one we haven't removed the s0 to be one we haven't removed the possibility that there is only zero and the S function loops around on itself so let's introduce a new rule to do that
[07:10] let's introduce a new rule to do that for every counting number Square s² = 0 for every counting number Square s² = 0 is false great now we have Z and s0 in is false great now we have Z and s0 in other words 0 and one though we've sort
[07:26] of shifted our looping problem across by one step since we can have s0 equal ss0 can we fix that let's introduce another axom to clean that particular
[07:41] problem up for every pair of counting numbers square and triangle if s Square numbers square and triangle if s Square = s triangle then square equals triangle basically two different numbers can't reach the same successor this stops any
[07:58] change arising from zero forming a loop because at some point in the chain coming up from zero the chain would have to join the loop that would create the situation where two different numbers would have
[08:13] the same successor however it is not perfect as it doesn't exclude a freestanding Loop that doesn't originate from zero let's introduce one last axium
[08:25] for that you know when I said that sometimes we have to lie in maths explainer videos on the internet to prevent them from getting too long and prevent them from getting too long and distracted by complex mintii lies told
[08:39] distracted by complex mintii lies told to students where you simplify a complex subject by leaving out the things that would distract or confuse this is one of the things I'm going to have to lie about for every property if zero has
[08:54] that property and assuming Square also has that property allows you to prove has that property allows you to prove that s squ also has that property then all counting numbers have that property this is the principle of induction and I
[09:09] this is the principle of induction and I love inductive proofs so very much this principle of induction will safely banish any non-standard construction of
[09:22] the counting numbers away we don't have to worry about any non-standard numbers to worry about any non-standard numbers in our counting numbers the principle of
[09:34] in our counting numbers the principle of induction as stated by me in this video induction as stated by me in this video is completely standard and completely is completely standard and completely unpro being incompatible with the logic
[09:46] unpro being incompatible with the logic we use for everyday or mathematics is just a minor technicality that you don't have to worry about please ignore anybody who says anything about second
[09:58] anybody who says anything about second order logic axom schemers or the lellan order logic axom schemers or the lellan skm scolum theorem these are unimportant distractions that are not worth your consideration the remaining piano axioms
[10:12] go on to Define addition multiplication and order with them in place no matter and order with them in place no matter how we construct counting so long as we how we construct counting so long as we follow the piano axioms they will behave
[10:27] in the same way if we have a counting system that is constructed from bushels of wheat it will behave in exactly the same way as a counting system same way as a counting system constructed of marks on clay tablets if
[10:42] you do addition and subtraction on the clay tablets it will correspond to the adding and removal of wheat in the stores this sympathy between different
[10:55] systems is super useful if you happen to be a society that grows wheat and stores it in a centralized Granary so we have finally defined the counting numbers but
[11:07] that really hasn't solved my problem has it like I said before if we have to give it like I said before if we have to give something an extra name that suggests that we need to distinguish it from something else there are more to numbers
[11:21] than just the counting numbers shocking I know the counting numbers are extended into the whole numbers which are also called the integers the holes are
[11:33] extended into the rationals and the rationals are extended into the still rationals are extended into the still badly named reels now for each of these number types we could do a bottom up construction of them
[11:48] construction of them alternatively we could do a top down axomic definition one of my earliest videos was me doing a top down axionic
[12:00] definition of the real numbers in part out of frustration for another YouTube video that Define the real numbers in a very poor way actually the fact that I
[12:12] very poor way actually the fact that I get aggravated over YouTube videos about get aggravated over YouTube videos about about math subjects kind of means that I don't think this is at all performative I think I just care a lot about math
[12:25] well that sort of solves the impostor syndrome how about the rest of it maybe a good definition of what a number is is the counting numbers and anything that the counting numbers and anything that is an extension on those numbers is also
[12:39] a number so by definition the holes the rationals and the reals they are all numbers and the reals can be extended into the complex numbers which most people agree are numbers excellent the complex numbers can be further extended
[12:57] into the querian which you might have come across if you've done any computer come across if you've done any computer 3D programming I don't see many references to the contan numbers compared to the complex numbers the next
[13:14] compared to the complex numbers the next extension to on the cantonian is the octonian and nobody thinks those are numbers perhaps if we can tell what's the difference between the cians and the complex numbers we will understand what
[13:28] is the secret source that makes numbers numbers well one big difference between the compans and the complex numbers is that compans and the complex numbers is that for the complex numbers multiplication
[13:43] for the complex numbers multiplication commutes that is square * triangle equals triangle * Square it doesn't matter what side of the time sign the complex number is on multiplication works the same
[13:57] for quorans it does matter what order you do your multiplication this is handy because there's another
[14:10] extension on top of the numbers called matrices we can add and multiply matrices we can add and multiply matrices but matrices also don't compute that it's that seems settled then any extension to the counting numbers is a
[14:26] extension to the counting numbers is a number so long as they commute oh dear there's still more running time on this video I suspect I'm missing something hold on didn't I make a joke about some sort of holy numbers the religious
[14:40] numbers of course are the cardinal numbers who are responsible for the election of the math Pope oh yes the cardinal numbers these along with the cardinal numbers these along with the AAL numbers extend the counting numbers
[14:55] beyond the limitations of the finite these are the trans finite numbers and since everybody calls them numbers I guess they must be that's fine since numbers they fit our current definition
[15:10] of what a number is and just let me check I'm sure that multiplication and addition commute for these numbers damn it ordinal addition and multiplication don't commute my definition doesn't work and
[15:26] honestly I thought I was going to end the video here with me left just as the video here with me left just as ignorant to what a number is as when I started and then I would have to work out how to transition smoothly
[15:44] into asking people to subscribe you know YouTube stuff but something happened while I was doing the research for this video and thinking it over I didn't find video and thinking it over I didn't find anyone with a good definition not one
[15:58] that satisfi me at least but I realize something what if I'm approaching the idea of classifying numbers from a wrong direction how about we take an approach that's a bit more taxonomic
[16:14] that's a bit more taxonomic taxonomy is the science of taxonomy is the science of classifying living things into various groups and various species the idea is that we will pick a holotype an example
[16:29] that we will pick a holotype an example of what a number is then we'll say that things that are like that holotype are numbers and things that are too different from the holotype are not numbers so I'm thinking we select the
[16:44] numbers so I'm thinking we select the counting numbers as our holotype specimen for numbers and decide our bounds of number numbers and decide our bounds of number isness based on that the holes the
[16:59] rationals and the reals they have a great deal in common with the counting numbers and everybody agrees these are numbers the complex numbers they commute
[17:12] like the counting numbers but you can't put complex numbers in an order but they have enough in common with the counting numbers to be considered numbers to be considered numbers the ordinal numbers don't
[17:27] numbers the ordinal numbers don't commute but they have order so they are commute but they have order so they are also number like enough to be also number like enough to be numbers the quorans don't commute and
[17:40] numbers the quorans don't commute and don't have order so we can feel safe thinking that they're not numbers so my personal definition of what a number is is the counting numbers as defined by the piano
[17:55] as defined by the piano aums and anything that behaves mostly like counting number however I'm fully aware of my limitations as a person is there something that's widely regarded as a number that my definition
[18:11] excludes is there something that isn't like a number that my definition like a number that my definition includes please tell me in the comments and while I'm doing this call to action could you also subscribe