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by Corry Shores
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[Mathematics, Calculus, Geometry, Entry Directory]
[Norman Wildberger, entry directory]
[Wildberger, Math Foundations, entry directory]
[The following is summary of Wildberger’s video lecture. You will find that he is a supremely talented teacher. Any mistakes are my own, as I am not a mathematician. Bracketed comments are my own and are not to be trusted. If I have any general commentary, it comes at the end.]
Norman J. Wildberger
Course Series
Math Foundations
Math Foundations A (1-79):
Arithmetic and Geometry
1
“What is a Number?”
Brief summary:
(1.1) Norman Wildberger is a research mathematician at the university of New South Wales in Sydney, Australia. He discovered rational trigonometry. This series will explore the foundations of modern mathematics in a new and better way. (1.2) There are three main aims to this series: {1} to expose some of the current weaknesses in set theory, analysis, geometry, and various other related areas of mathematics; {2} to create a framework that makes sense, that does not rely on authority or unsubstantiated axiomatic systems, one that any person can understand if they start from the beginning; {3} to provide a resource for teachers to help them design their curriculum. (1.3) Wildberger will not teach a course in mathematics where you build elaborate theorems but rather he will lay foundations. So he will examine various topics and determine what the essential notions and main definitions are and how they fit into place correctly. The overall test for if this is done the right way is if all the parts fit together in their natural way. (1.4) There are three main principles to this series: {1} we will start at the beginning (we do not assume we have sophisticated mathematical knowledge), {2} we will keep things simple and natural (we will connect with the real world), and {3} we will keep an open mind (we will not accept orthodoxy and we acknowledge that we may need to rethink previous ideas later on.) (1.5) We start with the most important objects in mathematics, namely, the natural numbers: 1, 2, 3, etc. The starting point for them is an empty page; we start mathematics with nothing.
(Image source. Wildberger, Math Foundations 1)
Next we introduce “something.” We write a stroke ‘|’
(Image source. Wildberger, Math Foundations 1)
It represents a single entity that we name “one”. The next idea is adding one to itself. We write “one and another one” as ‘||’, which makes a number that we call “two.” Then we can iterate this adding one to what we already have to obtain a sequence of objects, with their given names.
(Image source. Wildberger, Math Foundations 1)
As we can see, we are not assuming the Arabic-Hindu notation system, because it is a much more sophisticated system than this. So at this point (the beginning) a natural number is a string of ones.
(Image source. Wildberger, Math Foundations 1)
Mathematical objects will all be based on natural numbers, so everything else we will do are based on them. (1.6) We will make some observations about natural numbers: {1} they form a sequence. They are naturally ordered. To each natural number we may associate the next one, that can be called the “successor”. So the successor of | is || (or ‘two’). The successor of || (or ‘two’) is ||| (or ‘three’).
(Image source. Wildberger, Math Foundations 1)
This is our first example of a mathematical operation. We take a number, and its successor is another number. (1.7) If we want to count things, for instance little toys, we do the following. For the first object, we draw one stroke. For the next, another stroke, and so on until we have exhausted all of them, thereby obtaining a total number. (1.8) Another elementary concept is relative size: which one is bigger and which is smaller? Suppose we have two stroke numbers, one above the other, and we want to know which one comes earlier in the sequence. To do this, we pair the ones in each sequence.
(Image source. Wildberger, Math Foundations 1)
The sequence that has unpaired strokes is the one that is larger than the other. This gives us the notion of ‘bigger than’. (1.9) We will represent a natural number by a letter, like ‘n’ or ‘m’.
(Image source. Wildberger, Math Foundations 1)
This allows us to say, for instance, that n = m whenever the 1’s in n can be paired up with those in m [without remainders in either].
(Image source. Wildberger, Math Foundations 1)
This gives us our notion of equality, which is when numbers are the same. We also have a notion of inequality. When we write n < m, that means that n comes before m in the sequence of natural numbers.
(Image source. Wildberger, Math Foundations 1)
The next step will be to work on arithmetical operations on natural numbers.
[Introduction to Wildberger and This Series]
[The Three Overall Aims of the Series]
[Method]
[Main Principles]
[The First Notion: Natural Numbers]
[The Successor Operation]
[Counting Things]
[Relative Size]
[Notation for Numbers, Equality, and Inequality]
Online Description [Quoting]:
The first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then algebra, then analysis (calculus) and will also treat so called set theory. It will have a lot of critical things to say once we get around to facing squarely up to the many logical weaknesses of modern pure mathematics. The series is meant to be viewed sequentially. We spend a lot more time and effort than usual on fundamental issues with number systems. If you are a more advanced student, or a fellow mathematician, then the first few dozen videos might be a bit slow. But they are none-the-less important!(Written by Wildberger, youtube page)
Summary
(Repeats the “brief summary” and “contents” above, but with video links)
[Introduction to Wildberger and This Series]
[Norman Wildberger is a research mathematician at the university of New South Wales in Sydney, Australia. He discovered rational trigonometry. This series will explore the foundations of modern mathematics in a new and better way.]
(00.04-next)
[ditto]
[The Three Overall Aims of the Series]
[There are three main aims to this series: {1} to expose some of the current weaknesses in set theory, analysis, geometry, and various other related areas of mathematics; {2} to create a framework that makes sense, that does not rely on authority or unsubstantiated axiomatic systems, one that any person can understand if they start from the beginning; {3} to provide a resource for teachers to help them design their curriculum.]
(00.28-next)
[ditto]
[Method]
[Wildberger will not teach a course in mathematics where you build elaborate theorems but rather he will lay foundations. So he will examine various topics and determine what the essential notions and main definitions are and how they fit into place correctly. The overall test for if this is done the right way is if all the parts fit together in their natural way.]
(1.31-next)
[ditto]
[Main Principles]
[There are three main principles to this series: {1} we will start at the beginning (we do not assume we have sophisticated mathematical knowledge), {2} we will keep things simple and natural (we will connect with the real world), and {3} we will keep an open mind (we will not accept orthodoxy and we acknowledge that we may need to rethink previous ideas later on.)]
(2.06-next)
[ditto]
[The First Notion: Natural Numbers]
[We start with the most important objects in mathematics, namely, the natural numbers: 1, 2, 3, etc. The starting point for them is an empty page; we start mathematics with nothing.
(Image source. Wildberger, Math Foundations 1)
Next we introduce “something.” We write a stroke ‘|’
(Image source. Wildberger, Math Foundations 1)
It represents a single entity that we name “one”. The next idea is adding one to itself. We write “one and another one” as ‘||’, which makes a number that we call “two.” Then we can iterate this adding one to what we already have to obtain a sequence of objects, with their given names.
(Image source. Wildberger, Math Foundations 1)
As we can see, we are not assuming the Arabic-Hindu notation system, because it is a much more sophisticated system than this. So at this point (the beginning) a natural number is a string of ones.
(Image source. Wildberger, Math Foundations 1)
Mathematical objects will all be based on natural numbers, so everything else we will do are based on them.]
(03.10-next)
[ditto]
[The Successor Operation]
[We will make some observations about natural numbers: {1} they form a sequence. They are naturally ordered. To each natural number we may associate the next one, that can be called the “successor”. So the successor of | is || (or ‘two’). The successor of || (or ‘two’) is ||| (or ‘three’).
(Image source. Wildberger, Math Foundations 1)
This is our first example of a mathematical operation. We take a number, and its successor is another number.]
(6.04-next)
[ditto]
[Counting Things]
[If we want to count things, for instance little toys, we do the following. For the first object, we draw one stroke. For the next, another stroke, and so on until we have exhausted all of them, thereby obtaining a total number.]
(06.43-next)
[ditto]
[Relative Size]
[Another elementary concept is relative size: which one is bigger and which is smaller? Suppose we have two stroke numbers, one above the other, and we want to know which one comes earlier in the sequence. To do this, we pair the ones in each sequence.
(Image source. Wildberger, Math Foundations 1)
The sequence that has unpaired strokes is the one that is larger than the other. This gives us the notion of ‘bigger than’.]
(07.41-next)
[ditto]
[Notation for Numbers, Equality, and Inequality]
[We will represent a natural number by a letter, like ‘n’ or ‘m’.
(Image source. Wildberger, Math Foundations 1)
This allows us to say, for instance, that n = m whenever the 1’s in n can be paired up with those in m [without remainders in either].
(Image source. Wildberger, Math Foundations 1)
This gives us our notion of equality, which is when numbers are the same. We also have a notion of inequality. When we write n < m, that means that n comes before m in the sequence of natural numbers.
(Image source. Wildberger, Math Foundations 1)
The next step will be to work on arithmetical operations on natural numbers.]
(08.31-next)
[ditto]
Let us ask, what are some of the conceptual foundations of these mathematical foundations? In the first place, we are seeking foundations. We thus believe in grounding and building from secure foundations. This seems to have two purposes for Wildberger. One is pedagogical. Students (myself included) learn better when beginning with intuitive, basic principles, and gradually and continuously working to more complex notions that are based ultimately in those initial principles. The second purpose seems to be (and this may become more evident as we continue and learn more of Wildberger’s critiques of contemporary mathematics) that this is in line with Wildberger’s philosophy of mathematics. Wildberger thinks that mathematics is something natural, intuitable, coherent, and tied to reality. He says,
I am interested in laying the foundations, but not on building elaborate theories. So I am going to go around, and we are going to look at various topics and ask ourselves, what are really the essential notions here? What are the main definitions? How [do] they fit into place correctly? When mathematics is done right, all the blocks do fit together really well. It is not artificial. Things really do work out. And this is going to be the test for us that we are really doing things in the right way. So our main principles that we will follow is that we are going to start right from the beginning. We are not going to assume that you already have a Ph.D in set theory and logic. We are not going to assume a lot of sophistication, a lot of jargon; we are going to start from the beginning. The first steps are usually the most important ones in any journey, and it is especially true when developing mathematics: you have to start from the beginning, and you have to start in a simply way. So we are going to keep things simple and natural. We are going to try to connect with the real world at all times.
(01.33-2.45 emphasis my own choosing)
One interesting thing here is that math is intuitable, but also tied to reality. It is not that reality teaches us math (with us being totally passive) or that we impose an artificial, mathematical structure of consciousness awkwardly onto the world. Rather, it seems to me, our mathematical intuitions develop in concord with our interactions with the real mathematical properties of the real, natural world. And just as mathematical entities, properties, or processes have a real coherence, so too can our mathematical intuitions develop to have that coherence, in concord with the real world in our interactions with it.
Now let us consider some other conceptual foundations, as we look at Wildberger’s account of the natural numbers. The first is that natural numbers exist and are intuitable. They are real, in that we really are counting real things in the world when we count them, and we are not counting imaginary entities in our head. And they are intuitable, meaning that we can readily form a concept of their natural numericity. Now suppose we question this, and we say, ‘there is no buffalo. Our imagination drew a line around the buffalo to isolate it from the herd. But we could have drawn a line around the herd, or around the savanna; or we could have drawn a line around the buffalo’s head, then its eye’s,’ and so on. In other words, on this basis we might believe that natural numericity is still not something real. This brings us to another foundational, metaphysical notion that we should first address, namely, multiplicity and divisibility on the one hand, and unicity and indivisibility on the other. That there is a real world is something we are presupposing. (There is no grounds for this, but I would appeal to something with regard to human humility. We should not take the equally groundless position that humans have the power to completely fashion the world. Given how we face our limitations at every turn as humans, it seems unreasonable to attribute to us a godlike power of total world-creativity. And as we will see in a moment, the fact that the world affects us (it modifies our bodies, it can shock us, hurt us even) means that it would seem not to be entirely within our own creative activities. Also, the world seems to have certain regularities that we “discover,” that were already there even before human knowledge of them. As Kant famously wrote:
[…] representations that have often followed or accompanied one another are finally associated with each other and thereby placed in a connection in accordance with which […] one of these representations brings about a transition of the mind to the other in accordance with a constant rule. This law of reproduction, however, presupposes that the appearances themselves are actually subject to such a rule, and that in the manifold of their representations an accompaniment or succession takes place according to certain rules [...]. If cinnabar were now red, now black, now light, now heavy, if a human being were now changed into this animal shape, now into that one, if on the longest day the land were covered now with fruits, now with ice and snow, then my empirical imagination would never even get the opportunity to think of heavy cinnabar on the occasion of the representation of the color red.
Kant, Kritik der reinen Vernunft, Erster Teil, A100-101, pp.163-164; Critique of Pure Reason, p.229.[Kritik der reinen Vernunft, Erster Teil. Werke Vol. 3. Edited by Wilhelm Weischedel. Darmstadt: Wissenschaftliche Buchgesellschaft, 1968.
Critique of Pure Reason. Edited by Paul Guyer and Allen W. Wood. Translated by Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press, 1998.]
As a Deleuzean, I would point to the irregularities of the world and say because they affect us, they are not our creation. As Deleuze writes at the beginning of Ch.5 of Difference and Repetiton:
Difference is not diversity. Diversity is given, but difference is that by which the given is given, that by which the given is given as diverse. Difference is not phenomenon but the noumenon closest to the phenomenon. It is therefore true that God makes the world by calculating, but his calculations never work out exactly [juste], and this inexactitude or injustice in the result, this irreducible inequality, forms the condition of the world. The world ‘happens’ while God calculates; if the calculation were exact, there would be no world. The world can be regarded as a ‘remainder’, and the real in the world understood in terms of fractional or even incommensurable numbers. Every phenomenon refers to an inequality by which it is conditioned. Every diversity and every change refers to a difference which is its sufficient reason. Everything which happens and everything which appears is correlated with orders of differences: differences of level, temperature, pressure, tension, potential, difference of intensity.
Deleuze, Différence et répétition, p.286; Difference and Repetition, p.222.
[Différence et répétition. Paris: Presses universitaires de France, 1968.
Difference and Repetition. Translated by Paul Patton. New York: Athlone, 1994.]
In other words, I would note Deleuze’s analysis of shocking sensations (as in his Francis Bacon book), where the world shocks us with its irregularities and unpredictabilities. And I would say that we could only have those shocks were there a real world independent of our internal operations. (I am also a dialetheist, and I regard otherness, like the internal/external relation I am using here, as non-exclusive (the one can include the other in part) and non-exhaustive (the otherness of the other is not all possible otherness. See for instance Routley and Routley’s “Negation and Contradiction”, especially sections 3.11, 6, and 7.)
So we are supposing there is a real world. And we also know that our minds can discern multiplicities of unities, where any unity can itself be regarded as a multiplicity (by division), and any multiplicity as a unity (by combination). This is something made evident by a phenomenological analysis of the structures and operations of our consciousness. But we are also making the metaphysical claim that this applies to the real world, and the claim specifically here is that the world is made of countable parts. That claim is not well-established. For this notion I appeal to the natural, biological world, which seems to perform its own counting operations. Take a flower seed, plant it, let if flower and go to seed, then plant one of those seeds, and keep repeating this process. We will probably find that each time it flowers, it has the same number of petals. Pick another flower and do the same thing, and we will probably find it has a different number of petals. We also find very obvious geometrical and numerical patterns of all kinds in many plants. Nature seems to be counting. It seems to generate things (like petals) that are consistently countable as one number, and other things generate other numbers of parts. Now, imagine that one of our flowers that normally has five petals mutates and instead has five hundred petals. I would think it would not be able to survive, as it might be weighed down to the ground and will use too many resources to sustain such an oversized flower. In other words, the number of petals has consequences for the plant. So countable numericity is something that natural things seem to have some kind of knowledge of and implement and also depend upon for their existence. So to make the metaphysical claim that “all is one” or “all is multiple,” and thus that there are no countable things, would not fit in well with the natural fact that a flower generates a certain number of petals and not another number of them (while at the same time other flowers do in fact have another number of petals), and that number is vital to its survival. Now of course over time plants evolve to generate different numbers of parts (at different geometrical arrangements). But the fact that one number would be better suited to its environmental conditions in some situation rather than another number only helps our argument that natural numbers are real things in the natural world. For, these numbers are “differences that make a difference” in a real way (see Bateson’s formulation). So we have covered the following conceptual foundations: that knowledge can be effectively built upon principle foundations (because it allows for step-wise progressive learning and conceptual coherence throughout); that mathematical ideas, like number, are intuitable (because phenomenology tells us our consciousness involves related multiplicity-unity structures); that there is a correspondence between our mathematical intuitions and the mathematical properties of the real world (because these mathematical properties in the real world are things that our mathematical intuitions can be shaped by in our thoughtful interactions with them); that there is a real world (because it has certain properties, namely, regularities and irregularities, that cannot be explained simply by appealing to human creativity, and also, it shocks us, meaning that it is not something internal to us or originating from us); and that the real world indeed does have certain mathematical properties, including natural numerical properties (because plants for instance almost certainly know about and implement numerical counting or quantities).
Let us now move on to Wildberger’s account of natural numbers to explore some it its conceptual foundations. A natural number is fundamentally composed of units, starting with a singular unit. We established above that nature seems to generate and rely upon such units that form numerical groupings, like five singular petals. So we need not ground that concept further. But we add something new here when we represent those numbers, in this case as a stroke. Here we must deal with another assumption, namely, that numerical representation is possible and correspondent with numericity in the real world. This might be supported by the fact that a certain reliable regularity is noticeable in how this representation works. We write five strokes, one for each flower petal. It goes to seed, then regenerates, and the number of petals will correspond to our five markings, and this seems to have no end of correspondence. At this point we are not making the claim that our representations of natural numericity, our five strokes, tell us everything about natural numericity as it is in the world. We can only at this point say that it reflects certain aspects of it. For instance, it seems to share the same aspects of total quantification. Our five stokes (in the mind, on paper, among our fingers, etc.) corresponds with a mental notion of a quantity five (being one more than four and one less than six, etc.), and five petals seems to have a quantitative reality corresponding with that same mental conception (this flower has one more petal than a different flower with four petals, and it has one less petal than a flower with six, etc.). And our next claim, in the next lesson, is that there are certain natural and intuitable properties of natural numbers that we can model (for instance, associative and commutative laws). So our symbolic numerical systems can correspond with certain properties and operations of real things with regard to their numericity. But for the five-petalled flower, are its five petals something for it like our five strokes on a piece of bark? Or is that an anthropomorphism, namely, we count with our fingers, and thus we make strokes to conceptualize, analogically, numerical quantities? It would seem that our human conception and representation of number is contaminated by our humanity, and so it seems more reasonable to say not that our representations are mirrors of natural mathematics but rather that they are expressive of certain commonalities. So to our above list of conceptual foundations, we may add: that our representations of natural numericity correspond in some important way with how that natural numericity is in the real world (because the regularities of our representations correspond with regularities in the world, and other mathematical properties of the real world are modellable in a similar correspondent way with our mathematical representations.)
There are also other conceptions that should be grounded, but I am not capable of doing so. One is the notion of “successor”. I here would appeal to the notion of time, which is often associated with succession. Each moment of our lives never holds still, but rather gives way to the next moment. And we see time’s operations in the natural world with its natural cycles (of life and death, of the seasons, etc.) Another idea is that the strokes can be represented by alphabetical letters, where the letters need not represent some known and determinate value (they rather have variable and sometimes indeterminate meanings). Here I would acknowledge that while we may not have an intuition of a variable and indeterminate number quantity, we might see this involving something like what Hume says about abstract notions. The main idea, in application to our notion of letters-as-numbers, would be that when we see n or m, our mind does not conceive necessarily the exact value it represents in some equation, if it has one, nor does it conceive every possible number whatsoever; rather, we may vaguely have some number or other in the back of our mind, all while being ready to have any other number whatsoever instead in the back of our mind. Hume writes, but with respect to words and abstract ideas:
When we have found a resemblance among several objects, that often occur to us, we apply the same name to all of them, whatever differences we may observe in the degrees of their quantity and quality, and whatever other differences may appear among them. After we have acquired a custom of this kind, the hearing of that name revives the idea of one of these objects, and makes the imagination conceive it with all its particular circumstances and proportions. But as the same word is suppos’d to have been frequently applied to other individuals, that are different in many respects from that idea, which is immediately present to the mind; the word not being able to revive the idea of all these individuals, only touches the soul, if I may be allow’d so to speak, and revives that custom, which we have acquir’d by surveying them. They are not really and in fact present to the mind, but only in power; nor do we draw them all out distinctly in the imagination, but keep ourselves in a readiness to survey any of them, as we may be prompted by a present design or necessity. The word raises up an individual idea, along with a certain custom; and that custom produces any other individual one, for which we may have occasion. But as the production of all the ideas, to which the name may be apply’d, is in most cases impossible, we abridge that work by a more partial consideration, and find but few inconveniences to arise in our reasoning from that abridgment.
[Hume, pp.20-21. A Treatise of Human Nature. Ed. L.A Selby-Bigge. Oxford: Clarendon Press, 1979.]
We also have other mathematical notions that I am unable to give good grounding for, namely, equality, inequality, and sequence by amount. That two numerical representations be equal is easy to establish, like with Wildberger’s pairing operation. But that equality be something natural is a different matter. (Deleuze for instance thinks reality is thoroughly, at its fundamental level, composed of inequality and difference.) If we appeal to the flower petal example from above, it is contaminated by our representations that mediate between real instances. I here suppose we might appeal to our notion of justice here for an origin of our intuition about quantitative equality. (It is found in other animals as well. See here or here.) But still that is a matter of human (or animal) judgment and not necessarily something that manifests naturally in the real world. I guess physical experiments using a scale could demonstrate numerical equality in the physical world. But that assumes that already the two things being counter-balanced are themselves identical. In other words, we can show that two stones, one in each balance pan, come out equal, but first we had to find two stones that weigh exactly the same, which is a determination that requires a scale. If we choose acorns or other biologically generated items, they also will have different weights most times. So is equality in fact something real? Or is it only evident in abstract formulations or conceptions that idealize situations and strip them of their reality? I cannot give any metaphysical grounds for equality. It may be a property that only abstract numbers can have, being represented symbolically, yet perhaps at best in reality all is inequality, but in some cases that inequality is so low as to take on a significance in some situation that is functionally equivalent in certain respects to abstract equality. For instance, were we to put two rocks on the balance scale, and the pans lay equal, we would note that a more precise measurement would show some degree of inequality between them, only this instrument we now use is not that sensitive, and that with the most sensitive instrument, we would find no absolute equalities in the world. So inequality to me seems more metaphysically foundational, and I would in my mind regard what corresponds in the real world to the abstract equality relation as being rather a significantly low degree of inequality, with abstract equality being a limit case that may ultimately be unattainable in the real world.
But with that being the case, making an ordered sequence of natural numbers, the last idea of the lecture, is not so hard to think foundationally and metaphysically about. We mentioned already the steady flow of time. On its account, for instance, trees make a new “ring” in their trunks each year. Suppose we plant three trees of the same species. After 10 years, we cut down the first one. It has 10 rings, and we keep a slice of the trunk. After 11 years we slice the trunk of the next tree. We count 11 rings. In the first place, we notice an ordered succession (we could have started in fact from 1 and built up to 10 rings). And we see that the 11 rings is quantitatively larger than the 10, because the slice is a little larger. A year later we cut the last tree, which counts 12, and we see it is larger than the 11 ring slice, which is larger than the 10. So on account of the ordered succession of time in the physical world, we can see that our symbolic notion of succession and ordered inequality is not a mere invention of our minds but is something manifestly real about the physical world. So our final list of conceptual foundations are: that the real world has numerical succession (as seen in the effects of time); that the real world has inequality (as seen in physical objects being in relations of imbalance with regard to their weight); that the real world has something that in certain situations is functionally correspondent to abstract equality (as seen in certain instruments demonstrating a relative balance of physical objects, although maybe never being absolutely identical with abstract equality but tending toward it as a limit case); and that numerical sequences of ordered inequality is also something found in the real world (as seen, again, in the effects of time, on phenomena like growth for instance).]
Wildberger, Norman J. (2009). “What is a number?”. Part 1 of the course series: Math Foundations. Video.
Youtube page for this video:
Course Youtube Playlist:
Math Foundations A (1-79)
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Math Foundations B (80-149)
https://www.youtube.com/playlist?list=PLIljB45xT85DpiADQOPth56AVC48SrPLc
Math Foundations C (150 - )
https://www.youtube.com/playlist?list=PLIljB45xT85AYIeGfDQwHM8i6PQEDnnTI
Norman J. Wildberger, youtube channel:
[njwildberger]
Insights into Mathematics
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