## 16 Jun 2019

### Wildberger (2) Math Foundations, 2: “Arithmetic with Numbers”, summary

[Search Blog Here. Index tabs are found at the bottom of the left column.]

[Central Entry Directory]

[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

2

“Arithmetic with Numbers” (Image source. Wildberger, Math Foundations 2)

Brief summary:

(2.1) Last week we conceived the natural numbers as discrete unities that increase by accretion (adding strokes to sets of strokes. See section 1.5). (2.2) We imagine a boy a hundred thousand years ago sent to a water hole, tasked with counting the animals there. He sees buffalo (represented as squares) and antelope (represented as triangles). The boy then uses the counting procedure from section 1.7 to count the totals of each animal. He does so by making scratches in a piece of bark, thereby fashioning our numerical series of strokes. He counts the buffalo and makes one set of scratches for them, and he counts the antelope and makes another set of scratches for them. The boy returns to his village and shows the people his count. But then they ask him to go back for a total count of all animals. Yet, the boy uses a technique to get the full count without going back to the water hole and repeating the whole counting procedure. The boy first copies the first set of strokes (for the buffalo), and he likewise copies the second set of strokes (for the antelope) right next to the first copied set of strokes. This is adding them. It indicates the total number of animals. (Image source. Wildberger, Math Foundations 2)

This is the operation of summation, specifically the sum of two numbers. It is defined as: the sum of numbers n and m is the combination (the putting together, concatenation) of the strings of ‘|’. We write this as n + m. (Image source. Wildberger, Math Foundations 2)

(2.4) Multiplication is more complicated. We consider a motivating example. We have three boxes, each with four circles. To total them, we combine four plus four plus four. (Image source. Wildberger, Math Foundations 2)

We think of it as ‘||| × ||||’, that is, as three groups of four. So we will replace each of the ones in the first string ‘|||’ with a copy of the second number ‘||||’. That gives us the longer string of twelve strokes: ‘||||||||||||’ (see image above). We thus define multiplication in the following way: The product of numbers n and m is the string formed by a copy of m for every 1 in n. It is written: n × m. (Image source. Wildberger, Math Foundations 2)

(2.5) There are three laws for multiplication. {1} The Commutative Law: n × m = m × m. {2} The Associative Law: (k × n) × m = k × (n × m). {3} The Identity Law (on the board but not spoken): n × 1 = n. (Image source. Wildberger, Math Foundations 2)

{4} The Distributive Laws. These connect addition and multiplication. {4a} k × (n + m) = (k × n) + (k × m). And we introduce an important notational convention here. We say that multiplication takes precedence over addition, and so we eliminate the parentheses: k × (n + m) = k × n + k × m. {4b} (k + n) × m = k × m + n × m. (Image source. Wildberger, Math Foundations 2)

The laws in these two section (including the laws of addition, section 2.3) are the most important ones in all of mathematics. “In fact, they are far more important than almost anything else. They are important because they are simple and because their effect in everyday life is so profound. All of mathematics uses these laws, directly or indirectly, all the time.” We need to understand why these laws are true in an intuitive and non-jargonistic way, in everyday language, using only the notation and conventions that we have so far established. Making such proofs, or reasons, for these laws, is our homework for the next session.

Contents

Online Description

2.1

[Review of Last Week]

2.2

[The Sum of Two Numbers]

2.3

2.4

[Multiplication]

2.5

[Laws for Multiplication]

General Commentary

Bibliography

Online Description [Quoting]:

We introduce the two basic operations on natural numbers: addition and multiplication. Then we state the main laws that they satisfy. This is a basic and fundamental fact about natural numbers; that we can combine them in these two different ways. A lot of arithmetic, and later algebra, comes down to the interaction between addition and multiplication!

Summary

(Repeats the “brief summary” and “contents” above, but with video links)

2.1

[Review of Last Week]

[Last week we conceived the natural numbers as discrete unities that increase by accretion (adding strokes to sets of strokes. See section 1.5). (Image source. Wildberger, Math Foundations 2)]

(0.08-next)

[ditto]

[contents]

2.2

[The Sum of Two Numbers]

[We imagine a boy a hundred thousand years ago sent to a water hole, tasked with counting the animals there. He sees buffalo (represented as squares) and antelope (represented as triangles). The boy then uses the counting procedure from section 1.7 to count the totals of each animal. He does so by making scratches in a piece of bark, thereby fashioning our numerical series of strokes. He counts the buffalo and makes one set of scratches for them, and he counts the antelope and makes another set of scratches for them. The boy returns to his village and shows the people his count. But then they ask him to go back for a total count of all animals. Yet, the boy uses a technique to get the full count without going back to the water hole and repeating the whole counting procedure. The boy first copies the first set of strokes (for the buffalo), and he likewise copies the second set of strokes (for the antelope) right next to the first copied set of strokes. This is adding them. It indicates the total number of animals. (Image source. Wildberger, Math Foundations 2)

This is the operation of summation, specifically the sum of two numbers. It is defined as: the sum of numbers n and m is the combination (the putting together, concatenation) of the strings of ‘|’. We write this as n + m. (Image source. Wildberger, Math Foundations 2)]

(1.03-next)

[ditto]

[contents]

2.3

[There are certain facts that we can discover about the addition operation, and these are the laws of addition. {1} We first see that it is commutative: n + m = m + n. We can see this in an illustration where we have five strokes and two strokes, which total seven strokes, and otherwise two strokes and five strokes, which also, in a readily visible way, totals the same amount of seven strokes. (Image source. Wildberger, Math Foundations 2)

{2} The second one is the associative law: (k + n) + m = k + (n + m). As we see, this one involves three values, k, m, and n. This gives us two options for making couples of addition. We can see, again, how this works with the strokes, because we neither increase no decrease the total number of stroke-objects, and so grouping them differently has no effect on the total sum. (Image source. Wildberger, Math Foundations 2)

{3} The third law is a minor one. We can write the successor operation s(n) as n + 1. (Image source. Wildberger, Math Foundations 2)]

(3.50-next)

[ditto]

[contents]

2.4

[Multiplication]

[Multiplication is more complicated. We consider a motivating example. We have three boxes, each with four circles. To total them, we combine four plus four plus four. (Image source. Wildberger, Math Foundations 2)

We think of it as ‘||| × ||||’, that is, as three groups of four. So we will replace each of the ones in the first string ‘|||’ with a copy of the second number ‘||||’. That gives us the longer string of twelve strokes: ‘||||||||||||’ (see image above). We thus define multiplication in the following way: The product of numbers n and m is the string formed by a copy of m for every 1 in n. It is written: n × m. (Image source. Wildberger, Math Foundations 2)]

(06.23-next)

[ditto]

[contents]

2.5

[Laws for Multiplication]

[There are three laws for multiplication. {1} The Commutative Law: n × m = m × m. {2} The Associative Law: (k × n) × m = k × (n × m). {3} The Identity Law (on the board but not spoken): n × 1 = n. (Image source. Wildberger, Math Foundations 2)

{4} The Distributive Laws. These connect addition and multiplication. {4a} k × (n + m) = (k × n) + (k × m). And we introduce an important notational convention here. We say that multiplication takes precedence over addition, and so we eliminate the parentheses: k × (n + m) = k × n + k × m. {4b} (k + n) × m = k × m + n × m. (Image source. Wildberger, Math Foundations 2)

The laws in these two section (including the laws of addition, section 2.3) are the most important ones in all of mathematics. “In fact, they are far more important than almost anything else. They are important because they are simple and because their effect in everyday life is so profound. All of mathematics uses these laws, directly or indirectly, all the time.” We need to understand why these laws are true in an intuitive and non-jargonistic way, in everyday language, using only the notation and conventions that we have so far established. Making such proofs, or reasons, for these laws, is our homework for the next session.]

(07.35-end)

[ditto]

[contents]

In the case of the laws of addition (2.3), we can see the value of Wildberger’s stroke-representation approach. For instance, for the commutative law (n + m = m + n), we can visually see the identity between the two sets of stroke additions. (Image source. Wildberger, Math Foundations 2)

This law seems in a sense to be a property of the line objects themselves, which could be easily discovered by playing with them in a spontaneous way. For instance, if we had a group of stroke-like objects, and we divided them into two groups, we would find that no matter how we made those divisions, we still in total will have the same number of objects, as none will have disappeared or come into existence out of nowhere. And more to the point, we will see that simply by moving the divided groups around changes nothing in the total number. This sort of obvious mode of demonstration would seem to work for the multiplication laws as well, and thus Wildberger is fulfilling his characterization of mathematics as something natural, intuitable, coherent, and tied to reality (see section 1.4 and the General Commentary to that first lesson.)

I would like to continue with some assumptions we discussed last time in the general commentary. We wanted to know the grounds of equality. And we said that in our intuition and observations of the world, there seems to be no such thing at least in an objective sense. However, we said that in certain situations, two things that are not equal can be unequal to a degree that in a particular context is negligible, and so for all practical purposes they are equal for us. Two objects on a balancing scale may show on the read-out to be the exact same weight, but on a more sensitive scale they would show to be a little off. But insofar as the first scale is sensitive enough for our needs, the two are minimally unequal enough to be treated as equal. I mention that, because we missed an important assumption in the last commentary that comes more to light with the buffalo/antelope example from section 2.2. Suppose the story went the other way. The boy’s task is to count the total animals first, which he does. He thus first ignores the differences between buffalo and antelope. But next, he is asked for a specific count of each animal type. Now he makes that distinction, giving strokes in different groupings for the different types of animals. Here, while the groupings are thought to be distinct, the items in each group are all homogeneously repetitions of strokes. Suppose further that the boy’s third task is to be more precise, and to give counts of all the male antelopes versus female, and all the male buffalo versus female. Now he will have four groups of strokes, noticing more differences. And this can be repeated. What we might observe here is the denuding of real things of their particularities and representing them with strokes. This is an assumption we should now consider, namely, that any particular object can be represented by a quantifying symbol that is homogeneous with the quantifying symbols given to any other object. The basis for this, it seems to me, is the widening of generalities of types. We can think of animals, more generally, living things, then physical things, and most generally of entities of any sort whatsoever, either existant or not. The question is whether or not this is a falsification of reality; for, there are no things either in the imagination or in the real world that have no particularizing properties of any sort. What is a thing with no properties whatsoever? There is no such thing. But what are the strokes representing the animals? They are things with no particularizing properties of any sort. Nonetheless, we know that the natural numbers exist in nature (see this entry on how they are exhibited in waves). At this moment, I am not sure where to go with this, but I will try to come back to it. Perhaps natural numbers in the natural state should be understood as, to give some possibilities: abstractions like our strokes, but in a cosmic consciousness that pervades all things; ratios between differences of physical quantity, somehow; structural relationships of some sort that can be understood in terms of numerical quantities but in themselves are simply natural structuring relationships of some sort; or, the physical world is like a giant computer, and somehow fundamentally all things are number or ratio, but on the phenomenal level they appear as non-numerical.]

Wildberger, Norman J. (2). “Arithmetic with Numbers”.  Part 2 of the course series:  Math Foundations. Video.

https://youtu.be/-96tlu_sShM

Math Foundations A (1-79)

Math Foundations B (80-149)

Math Foundations C (150 - )

[njwildberger]

Insights into Mathematics

## 12 Jun 2019

### Wildberger (1) Math Foundations, 1: “What is a Number?”, summary

[Search Blog Here. Index tabs are found at the bottom of the left column.]

[Central Entry Directory]

[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.

Contents

Online Description

1.1

[Introduction to Wildberger and This Series]

1.2

[The Three Overall Aims of the Series]

1.3

[Method]

1.4

[Main Principles]

1.5

[The First Notion: Natural Numbers]

1.6

[The Successor Operation]

1.7

[Counting Things]

1.8

[Relative Size]

1.9

[Notation for Numbers, Equality, and Inequality]

General Commentary

Bibliography

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!

Summary

(Repeats the “brief summary” and “contents” above, but with video links)

1.1

[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]

[contents]

1.2

[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]

[contents]

1.3

[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]

[contents]

1.4

[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]

[contents]

1.5

[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]

[contents]

1.6

[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]

[contents]

1.7

[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]

[contents]

1.8

[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]

[contents]

1.9

[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]

[contents]

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.

https://youtu.be/91c5Ti6Ddio

Math Foundations A (1-79)

Math Foundations B (80-149)

Math Foundations C (150 - )

[njwildberger]

Insights into Mathematics

## 9 Jun 2019

### Heyting (4) “G. F. C. Griss and His Negationless Intuitionistic Mathematics.” Section 4, “[Griss’ Negationless Mathematics and Real Numbers]”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Heyting, entry directory]

[Heyting’s ”G. F. C. Griss and His Negationless Intuitionistic Mathematics”, entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are possibly mistaken and inelegantly articulated. Bracketed comments are my own, and the section enumerations follow the paragraph divisions. Proofreading is incomplete, so please forgive my mistakes.]

[Note, this post comes at the end of a series on negationless intuitionistic mathematics, and it in particular synthesizes all the other ones. See the collected brief summaries on this topic by Griss and Heyting.]

Summary of

Arend Heyting

”G. F. C. Griss and His Negationless Intuitionistic Mathematics”

4

“[Griss’ Negationless Mathematics and Real Numbers]”

Brief summary:

__(4)__Griss, as a philosopher and mathematician, thought both theoretically about a negationless intuitionistic mathematics, and also constructed it formally. Griss constructed the natural numbers using a positive notion of difference (namely, being in a subset that is complementary to the other subset that contains all the rest of the numbers in the larger, whole set). And we can also determine when natural numbers are equal. [Their equality can be established, not in the negational way of saying that it is impossible that they are unequal, but rather in the negationless way of saying that they share differences to exactly the same other numbers.

If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.2,  p.1132)

a c for each c b a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3,  p.1132) ] Rational numbers are defined as pairs of natural numbers. But real numbers are more complicated. They are defined as sequences of approximating intervals that converge upon a value. (They are Cauchy sequences of rational numbers that, as they go further down their sequence, form intervals between one another that eventually become arbitrarily small and convergent upon a particular value, which is the real number value expressed by that convergent series. (Image from: Norman Wildberger, Math Foudations 111) (Image source: wiki)

) (Following Edna Kramer, we could also call to mind another sort of narrowing, approximating intervals that are probably more intuitive for us, namely, the decimal expansion of a real number with non-terminating (and possibly non-repeating) decimals. Each additional decimal will have yet another coming after it. For instance, consider such a number that begins with 2 and will next have 2.6. (Kramer, Nature and Growth of Modern Mathematics, section 2.x.1, p.34, boldface and underlining are mine)

That means previously at 2, it was really an interval spanning 2 and 3, because more precisely it will be at 2.6, which falls between 2 and 3. And after 2.6 is 2.63. So in fact, at 2.6 it is an interval between 2.6 and 2.7. Since yet another interval comes after the .63, that means it was an interval between 2.63 and 2.64. And so on. While this may not be a Cauchy sequence, it at least gives us the image of a series of narrowing, approximating intervals that ultimately converge upon a real number value, in a way that we are more familiar with.] Heyting calls such a series expressing a real number a “real number-generator”. When two such number-generators have terms (and approximating intervals) that all overlap, then they are the same. (Note: we are not yet at Griss’ definition of the equality of real numbers.) Next we will see Griss’ negationless conception of the inequality of two real numbers. The negational way that Griss rejects is to say that two real numbers are unequal if it is impossible that they are equal. For, this uses the negational notion of “impossibility” (and probably a reductio method of proof). Instead, the notion of inequality is understood positively as a distance or gap between them (between their approximating intervals). This apartness relation is symbolized with ‘⧣’. And it is defined in the following way: “two real numbers, defined by the number- generators a = {an} and b = {bn} are apart from each other (ab) if for some n, an and bn are separated intervals” (Heyting 94). [Griss in one place words it: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another” (Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128). In other words, while two close real numbers may have many of their approximating intervals sharing common ‘space’, at some point along the sequence, the intervals will occupy space outside the other. ] So that defines the inequality of real numbers in a negationless, intuitionistic mathematics. But, the equality of two real numbers cannot then be defined negatively as the impossibility of their being apart. Instead, Griss defines the equality of two real numbers as their sharing distances to all the other real numbers. In Heyting’s wording: “if every real number c that is apart from a is also apart from b, then a = b” (94). [ ]

Contents

4

[Griss’ Negationless Mathematics and Real Numbers]

Bibliography

Summary

4

[Griss’ Negationless Mathematics and Real Numbers]

[Griss, as a philosopher and mathematician, thought both theoretically about a negationless intuitionistic mathematics, and also constructed it formally. Griss constructed the natural numbers using a positive notion of difference (namely, being in a subset that is complementary to the other subset that contains all the rest of the numbers in the larger, whole set). Rational numbers are defined as pairs of natural numbers. But real numbers are more complicated. They are defined as sequences of approximating intervals that converge upon a value. (They are Cauchy series of rational numbers that, as they go further down their sequence, form intervals between one another that eventually become arbitrarily small and convergent upon a particular value, which is the real number value expressed by that convergent series.) Heyting calls such a series expressing a real number a “real number-generator”. When two such number-generators have terms (and approximating intervals) that all overlap, then they are the same. (We are not yet at Griss’ definition of the equality of real numbers.) Next we will see Griss conception of the inequality of two real numbers. The negational way that Griss rejects is to say that two real numbers are unequal if it is impossible that they are equal. For, this uses the negational notion of “impossibility” (and probably a reductio method of proof). Instead, the notion of inequality is understood positively as a distance or gap between them (between their approximating intervals). This apartness relation is symbolized with ‘⧣’. And it is defined in the following way: “two real numbers, defined by the number- generators a = {an} and b = {bn} are apart from each other (ab) if for some n, an and bn are separated intervals” (Heyting 94). [Griss in one place words it: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another” (Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128).] So that defines the inequality of real numbers in a negationless, intuitionistic mathematics. But, the equality of two real numbers cannot then be defined negatively as the impossibility of their being apart. Instead, Griss defines the equality of two real numbers as their sharing distances to all the other real numbers. In Heyting’s wording: “if every real number c that is apart from a is also apart from b, then a = b” (94).]

[Oftentimes philosophical ideas encounter problems when applied to concrete problems. Griss, however, did both the philosophical and applicational work, being both a philosopher and mathematician: “after a short philosophical introduction he begins the construction of mathematics” (93). His main aim is to “to find a substitute for reasonings which involve negation; simply banishing these he would leave but insignificant ruins.” [Recall from section 0.2 of “Negationless Intuitionistic Mathematics, I” that Griss writes:

On philosophic grounds I think the use of the negation in intuitionistic mathematics has to be rejected. Proving that something is not right, i.e. proving the incorrectness of a supposition, is no intuitive method. For one cannot have a clear conception of a supposition that eventually proves to be a mistake. Only construction without the use of negation has some sense in intuitionistic mathematics.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.2, p.1127)

] For this, he needs to see how difference operates for different kinds of numbers. Heyting begins with Griss’ non-negational intuitionistic formulation of the natural numbers. Let us quote Heyting first, then we will examine Griss’ relevant texts:

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

(Heyting 93)

First we need to be clear about what constitutes a negative mathematical formulation, conception, or proof. As we will see, it is one that involves a conception of negating a verb/predicate or whole proposition. In section 0 of Griss’ “Negationless Intuitionistic Mathematics, I” he fashions two sorts of mathematical proofs, one using negation and another that is negationless (section 0.5). Both will show that a certain line bisecting a triangle (constructed according to certain proportional conditions) will be parallel to one of the sides of the triangle. The first proof is negational, because it uses a reductio argument. It assumes that this bisecting line is not parallel:

If DE was not parallel with AB, ...

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.5, p.1128)

Then it finds a contradiction, from which we infer that the lines are parallel. But for the negationless proof, we first construct the line in question. We next construct another line that we know to in fact be parallel to the triangle side. Finally we show that this parallel line must necessarily be identical to the line in question. Since the second line is parallel, and since it is identical to the line in question, then the line in question is therefore parallel (section 0.5). Furthermore, the mathematical notion of parallel lines can be given either a negative or positive definition (section 0.6). The negative definition of parallel lines is:

parallel lines (in a plane) are lines which do not intersect

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

Here we see that the predicate, “intersect” is negated as “do not intersect.” But the positive definition would be:

parallel lines are such lines, that any point of one of them differs from any point of the other one. And this, again, presupposes a positive definition (i.e. a definition without negation) of difference relative to points.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128, boldface and underlining mine, italics in the original)

[But how we get a positive notion of difference is something we deal with in a short while.] Again we see that positive notion of difference in the following reformulation:

Which rational numbers satisfy x– 2   = 0? The answer must be: No rational number satisfies. The question has been put in the wrong way. The fact is that x– 2 differs positively from zero for every rational number.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

We will return to the next example later, because it uses notions we will work toward progressively. Before we move on, we should note one of Heyting’s explanations for negation, coming from his Intuitionism: An Introduction. He begins with the following negational yet intuitionistic propositions:

If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a b.

Theorem 1.   If a b is contradictory, then a = b [L. E. J. Brouwer 1925, p . 254].

(17)

BROUWER, L. E. J.

1925. Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht deutsch. Math. Ver. 33, p. 251–256.

(124)

(Heyting, Intuitionism: An Introduction, sections 2.2.2.1 and 2.2.2.2, p.17 [Bib p.124])

Heyting then clarifies that there are two sorts of negation in this intuitionistic context. One is simply de facto negation, meaning that we do not yet have a proof for something. This is just lacking a proof. (On this “weaker” sort of negation, see Mancosu and van Stigt, From Brouwer to Hilbert, section 1.5.1.5.)  The second kind is a de jure negation, which means you have a proof from which you can infer that some proposition is false. This is a disproof. (On this sort of stronger “Brouwer negation,” see see Mancosu and van Stigt, From Brouwer to Hilbert, sections 1.5.1.3 and 1.5.1.4.) But it is important to note that such a disproof is not formed by means of a reductio argument where the proposition is negated. So it is not like the reductio argument we noted above with the parallel dividing line of the triangle in Griss’ example. But in the non-negational case, after establishing that the line is parallel, you could then say, perhaps, that this constructed positive proof can serve to disprove that they are not non-parallel. As Mancosu and van Stigt explain in section 4.1.3 of From Brouwer to Hilbert:

“Intuitionist Splitting” is in fact such an exercise of creating new words, in this case words expressing the various relations between points and between points and species of points. In line with his own rules of correct logical practice, Brouwer starts from his concepts of mathematical truth and absurdity (i.e., proven impossibility), resulting immediately in the inapplicability of the Principle of the Excluded Middle and of what he calls “The Principle of Reciprocity of Complementary Species,” which asserts the equivalence of truth and double negation. He replaces the latter principle by a restricted form of complementarity: “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth.”

(Mancosu and van Stigt, From Brouwer to Hilbert, section 4.1.3, p.276)

And from section 4.2.1:

In particular the excluded middle and the principle of double negation were singled out as especially problematic. By contrast, Brouwer remarked that the intuitionist accepts the following principles: A → ¬¬A

(Mancosu and van Stigt, From Brouwer to Hilbert, section 4.2.1, p.274)

Thus Heyting writes:

Strictly speaking, we must well distinguish the use of “not” in mathematics from that in explanations which are not mathematical, but are expressed in ordinary language. In mathematical assertions no ambiguity can arise: “not” has always the strict meaning. “The proposition p is not true”, or “the proposition p is false” means “If we suppose the truth of p, we are led to a contradiction”. But if we say that the number-generator ρ which I defined a few moments ago is not rational, this is not meant as a mathematical assertion, but as a statement about a matter of facts; I mean by it that as yet no proof for the rationality of ρ has been given. As it is not always easy to see whether a sentence is meant as a mathematical assertion or as a statement about the present state of our knowledge, it is necessary to be careful about the formulation of such sentences. Where there is some danger of ambiguity, we express the mathematical negation by such expressions as “it is impossible that”, “it is false that”, “it cannot be”, etc., while the factual negation is expressed by “we have no right to assert that”, “nobody knows that”, etc.

(18)

There is a criterion by which we are able to recognize mathe- | matical assertions as such. Every mathematical assertion can be expressed in the form: “I have effected the construction A in my mind”. The mathematical negation of this assertion can be expressed as “I have effected in my mind a construction B, which deduces a contradiction from the supposition that the construction A were brought to an end”, which is again of the same form. On the contrary, the factual negation of the first assertion is: “I have not effected the construction A in my mind”; this statement has not the form of a mathematical assertion.

(Heyting, Intuitionism: An Introduction, sections 2.2.2.9 and 2.2.2.10, pp.18-19)

So to be clear, Griss thinks that all proofs must be positively constructive and all properties must be positively conceived and stated. This matter of positive conceptions we turn to now, and it brings us to the part of our current Heyting text we left of at a while ago, namely, the part reading,

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

(Heyting 93)

Griss writes in “Negationless Intuitionistic Mathematics, I”:

To construct negationless mathematics one must begin with the elements and a positive definition of difference must be given instead of a negative one (ex. 1 and 3).

But even from a general intuitionistic point of view a positive construction of the theory of natural numbers must be given: one cannot define 2 is not equal to 1 (i.e. it is impossible that 2 and 1 are equal), for from this one could never conclude that 2 and 1 differ positively.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.10, p.1130, boldface mine, italics in the original)

So we are first going to define the natural numbers in a positive, constructive way. But this will require a non-negational notion of difference. We will see how Griss does this also with the real numbers, and Deleuze uses Heyting’s formulation of it in Difference and Repetition. Griss begins with two primitive notions, being identical and being distinguishable (see section 3.2 of Griss’ “Logic of Negationless Intuitionistic Mathematics”). We need a positive definition of difference to define the natural numbers, because each one needs to be different from the others. It will be based on distinguishability, which is not defined, as we just noted. However, it will be given a conceptual formation and precise mathematical  formulation using notions of sets or “species.” We begin by imagining a selfsame object.

Imagine an object, e.g. 1. It remains the same, 1 is the same as 1, in formula 1 = 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.1, p.1131)

Next we imagine another object that is distinguishable from the first one. We call it 2.

Imagine another object, remaining the same, and distinguishable 4) from 1. e.g. 2; 2 = 2; 1 and 2 are distinguishable (from one another), in formula 1 ≠ 2, 2 ≠ 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.2, p.1131)

They are distinguishable from one another, and they form a set. So if we can distinguish one of them from 1, then it is 2, and vice versa.

They form the set {1, 2}; 1 and 2 belong to the set. If conversely an object belongs to this set, it is 1 or 2. If it is distinguishable from 1, it is 2; if it is distinguishable from 2, it is 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.3, p.1131)

We repeat this for 3, giving us the set {1, 2, 3} (see section 1.1.5). What is important here is that we can now regard this set as being made of complementary sets. We say that if an item is distinguishable from each element in the set {1, 2}, then it is 3, etc.

They form the set {1, 2, 3}. If an element belongs to {1, 2, 3}, it belongs to 1, 2 or it is 3. If it is distinguishable from each element of {1, 2}, it is 3; if it is distinguishable from 3, it is an element of {1, 2}.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.5, p.1131)

We can keep adding members, going up to some number n: {1, 2, ... , n}, and we can always imagine an additional n :

If, in this way, we have proceeded to {1, 2, …, n}, we can, again, imagine an element n′, remaining the same, n′ = n′, and distinguishable from each element p of {1, 2, ... , n}, in formula n′p, pn′.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.6, p.1131)

They form the set {1, 2, …, n′}. If an element belongs to {1, 2, …, n′}, it belongs to {1, 2, ... , n} or it is n′. If it is distinguishable from each element of {1, 2, ... , n}, it is n′; if it is distinguishable from n′, it is an element of {1, 2, ... , n}.

(1131)

At this point we need to emphasize very strongly that one element of this intuitionistic rejection of negation is a disjunctive sort of exclusion. Let us move to Griss’ “Negationless Intuitionistic Mathematics, II”, where he addresses this. But we need to modify our notation a little, although it will yield the same structure above, namely, we will call the set of natural numbers {1, 2, …, n} as En (1, 2, ..., n), and {1, 2, …, n′} as En′ (1, 2, ..., n′). I will begin with the full quote, and then we will analyze the key parts, as we need this for filling out our notion of negation:

ad §1.1.     After the introduction of the natural numbers 1, 2, 3 the | natural number n′ next to the natural number n was introduced by means of induction as follows:

“If, in this way, we have proceeded to En (1, 2, ..., n), we can again imagine an element n′, remaining the same, n′ = n′, and distinguishable from each element p of En (1, 2, ..., n), in formula n′ ≠ p, p ≠ n′. They form the set En′ (1, 2, ..., n′).”

En′ is called the sum of En and n′, in other words: An element of En′ belongs to En or is n′. In this way the disjunction is defined in a particular case. It is evident the disjunction a or b in the usual meaning (the assertion a is true or the assertion b is true), does not occur in negationless mathematics, because there is no question of assertions that are not true. In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.

(Griss, “Negationless Intuitionistic Mathematics, II”, section 1.0.3, pp.456-457)

What we have seen in these Griss texts is that what makes a number be unique is if it can be distinguished from the set of all the remaining numbers, within the larger set it is a part of. And for the larger whole set to be divided in this way, we need a disjunction that says a number is either in the one or in the other. Again:

En′ is called the sum of En and n′, in other words: An element of En′ belongs to En or is n′. In this way the disjunction is defined in a particular case.

Now let us narrow in on the key passage (again):

It is evident the disjunction a or b in the usual meaning (the assertion a is true or the assertion b is true), does not occur in negationless mathematics, because there is no question of assertions that are not true.

The usual meaning of ‘a or b’ is: either a is true or b is true. That much is fine. But he claims that this involves a conception of untrue assertions. The problem is that he never mention untrue assertions or falsity in that formulation. What might it be? It would seem to be a sort of disjunctive syllogism where the untruth of a allows us to know the truth of b. He does not say this. But as we will see with his own definition, he will convert this disjunction into a conjunction of mutually affirmative conjuncts, even though it articulates a distinction between the conjoined parts. He writes (again):

In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.

So to be clear, disjunction here is not understood in the classical, negational sense as meaning that the falsity of one disjunct entails the truth of the other. [In intuitionistic negationless mathematics, we cannot conceive a falsity and we cannot blindly assert it. So perhaps  this clarifies two reasons for the intuitionistic prohibition of the principle of excluded middle, at least in this negationless mathematics context. The first is that we can have a proposition and its negation both being false (in the weak sense), if it has not yet been proven. And so the falsity of one cannot be seen as exclusive to the falsity of the other. The second is that we cannot conceive a (strong) falsity in the first place. We can only infer it by means of constructive, positive proofs. Also note this quotation from Griss:

In 1947 Prof. L. E. J. BROUWER gave a formulation of the directives of intuitionistic mathematics 2). It is remarkable that negation does not occur in an explicit way, so one might be inclined to believe negationless mathematics to be a consequence of this formulation. The notion of species, however, is introduced in this way (translated from the Dutch text): “Finally in this construction of mathematics at any stage properties that can be supposed to hold for mathematical conceivabilities already obtained are allowed to be added as new mathematical conceivabilities under the name of species”. By this formulation it is possible that there are properties that can be supposed to hold for mathematical conceivabilities already obtained but that are not known to be true. With it negation and null-species are introduced simultaneously but at the cost of evidence. Whatever are the properties that can be supposed? What other criterion could there be than ‘to hold for mathematical conceivabilities already obtained’? In the definition of the notion of species the words “can be supposed” should be replaced by “are known”. One should restrict oneself in intuitionistic mathematics to mathematical conceivabilities and properties of those mathematical conceivabilities and one should not make suppositions of which one does not know whether it is possible to fulfil them. (The well-known turn in mathematics: “Suppose ABC to be rectangular” seems to be a supposition, but mostly means: “Consider a rectangular triangle ABC”).

(457)

2) L. E. J. BROUWER, Richtlijnen der intuïtionistische wiskunde. Proc. Kon. Ned. Akad. v. Wetensch., 50, (1947).

(Griss, “Negationless Intuitionistic Mathematics, II,” section 1.0.3, p.457, italics in the original)

There Griss explains why there can be no null-set or property. We cannot conceive of a property that no thing can have.] We thereby can understand disjunction as a conjunction of terms that are in different sets or that thus have different properties. However, the exclusive element here is built into the notion of complementarity, but it is not initially conceived as such. First we say that a is in one subset and thus has some property, then we say b is in another subset and thus has some other property, and finally, the fact that one plus the other makes the larger set entails the members of one not being in the other. We did not begin with that exclusionary notion or definition of complementarity, although we arrived upon it. We will next look at how equality can be defined in a negationless way. The negational definition for equality says,

If it is impossible, that a is not the same as b, then a is the same as b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3,  p.1132)

[Note that there is a similar one for inequality:

a est différent de b, (a b), signifie, dans la terminologie de BROUWER, que a = b est impossible.

(Heyting, Les fondements des mathématiques, section 5.3.1.1, p.24)

] Griss reformulates the above negational definition for equality into the following negationless kind:

If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.2,  p.1132)

a c for each c b a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3,  p.1132)

So we begin with two unidentified numbers, and we want to know if they are equal. If they are both distinguishable (unequal) from precisely every other same number in the set, then they are equal to one another. In other words, two numbers are equal if they share the same differences or distinguishability relations to the other members. This means that they stand outside the set of all the other numbers but that one. If we think of a simplistic case where we have three numbers, with 1 and 3 being included, a and b would be equal if they are each different from those other terms. So this completes the section on natural numbers. We see now that in negationless intuitionistic mathematics, natural numbers can be constructed member-by-member in a positive way on the basis of a distinguishability from the all other natural numbers already in the set. And, this is not a matter of not being in the other set, but rather of being in the additional set, which, when combined with the first, completes the whole set. Furthermore, properties like equality can be defined without the notion of an impossibility of it being otherwise but rather as an affirmation of all their shared differences or distinguishabilities to the other numbers. Let us return to the text at hand, picking up where we left off:

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

(93)

But matters are more difficult for rational numbers. Yet, we will need this conception if we want to fully grasp Deleuze’s interest in negationless intuitionistic mathematics.

For real numbers the case is different. A real number is defined by a convergent, contracting sequence of rational intervals; for the sake of brevity I shall call such a sequence a number-generator. Two number-generators a = {an} and b = {bn} coincide, if an and bn overlap for every n. Coinciding number-generators define the same real number;

(93)

We will need to unpack this. The first concept is rational numbers, which are ones that can be expressed as an integer over an integer. (See Wildberger, Math Foundations 13, 01.30.) [Perhaps this is what Heyting means when he says that they “are defined as pairs of natural numbers”, but I am not sure.] The next concept is real number, which will get a special definition here. [For a discussion of the conventional definitions, see Wildberger’s Math Foundations 115, 02.50. Recall that real numbers can be expressed in decimal form, whether it be terminating or not, and repeating or not; that real numbers include the rational and the irrational; and that real numbers are ones that can be understood, as wikipedia says, as a “value of a continuous quantity that can represent a distance along a line.”] Heyting defines a real number as “a convergent, contracting sequence of rational intervals”, and he construes them as “real number-generators” (or just “number-generators” in this context). So what is a  number-generator? In Heyting’s Intuitionism: An Introduction, section 2.2.1.4, he says that they are Cauchy sequences of rational numbers.

Definition  1. A Cauchy sequence of rational numbers is a real number-generator.

(Heyting, Intuitionism: An Introduction, section 2.2.1.4, p.16)

What is a Cauchy sequence? Here is Heyting’s definition:

A sequence {an} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k), such that |an+pan| < 1/k for every natural number p.

(Heyting, Intuitionism: An Introduction, section 2.2.1.2, p.16)

This is quite complex. Generally speaking, a Cauchy sequence is one with a series of rational numbers that progressively tend toward an ultimate value, with the gap between successive numbers narrowing upon that ultimate value. Wildberger, in Math Foundations 111.6, shows this gradual, interchanging convergence of the values with this diagram: (Image from: Norman Wildberger, Math Foudations 111)

The green line is the value that the series of rationals are tending toward. The idea was that no matter how small an interval you choose, you will be able to find a place in the sequence after which the gaps between successive values (the space above and below the green line) will be less than that arbitrarily small interval. This implies that it is always moving toward some specific value (the green line) that it converges upon. So let us look again at the more formal definition again:

‘A sequence {an} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k), such that |an+pan| < 1/k for every natural number p.’

(Heyting, Intuitionism: An Introduction, section 2.2.1.2, p.16)

Here, the 1/k is the arbitrarily small interval. The larger the k value, the smaller the interval. The definition here says that no matter how large the k value (and thus no matter how small the interval), there will be some point along the sequence, some nth term, after which no matter what further point you select (no matter what p), the difference between successive terms will be smaller than that arbitrarily small interval. We see that narrowing of values also in this diagram from wikipedia of a Cauchy sequence: (Image source: wiki)

We might also think of this narrowing of intervals in a related (but probably not equivalent) way as a progressive determination of intervals in a decimal expansion, which Edna Kramer does in Nature and Growth of Modern Mathematics, section 2.x.1. There we said that a real number can be considered as a series of approximating intervals, getting smaller and smaller, and converging upon a particular point on the number line (and thus to an exact value), even if the decimals are non-terminating and non-repeating. Each new decimal, when taken along with the decimal value of one higher, creates an interval, with each one being nested within the prior one and all shrinking down to a particular point. She writes:

2.6314 ... . The decimal gives us a sequence of rational approximations to the real number, namely, 2, 2.6, 2.63, 2.631, 2.6314, ... . In other words, the first approximation in the sequence places the real number in the interval (2, 3), and then 2.6 gives the approximating interval (2.6, 2.7), etc. Thus we have the sequence of nested intervals, (2, 3), {2.6, 2.7), (2.63, 2.64), ... , illustrated in Figure 2.9. The adjective nested describes the fact that each interval lies within the preceding one. We observe also that the lengths of successive intervals are. 1, 0.1, 0.01, 0.001, 0.0001, ... . Since we are considering a nonterminating decimal, the nest of intervals will ultimately contain an interval of length 0.000 000 001 and then there will be still smaller intervals, so that interval length shrinks toward zero. As the innermost intervals get smaller and smaller, one can imagine their bounding walls approaching collision or, at any rate, getting close enough to “trap” a point of the number line. It is postulated, that is, assumed, that there is a unique point contained in all intervals of the nest. If there is such a point, we see that it must be unique, for if there were another distinct point, it would be separated from the first by some distance, 0.000 01, say. But ultimately some interval of the nest will be smaller than that number, and the first point must be contained in that very small interval. Then the second point would be too far away to be inside the interval and hence would not be contained in every interval of the nest. Since every nonterminating decimal will give rise to a sequence of nested intervals like the one described, there will always be a unique point of the number line corresponding to every real number.

(Kramer, Nature and Growth of Modern Mathematics, section 2.x.1, p.34, boldface and underlining are mine)

So a real number-generator is a real number as defined as being a Cauchy sequence of rationals, in other words, as a series of approximating intervals narrowing down and converging upon a singular value, even if the decimal expansion is non-terminating and non-repeating. Let us return to our current text and pick up on the next notion:

Two number-generators a = {an} and b = {bn} coincide, if an and bn overlap for every n. Coinciding number-generators define the same real number

(Heyting 93)

So here we see that two number-generators (two real numbers) coincide if the series “overlaps” for every term in the series. This brings us to some complexities, but we will simplify them eventually. In Heyting’s Intuitionism: An Introduction, section 2.2.1.5, he defines the identity of number-generators in a similar way:

Two number-generators a ≡ {an} and b ≡ {bn} are identical, if an = bn for every n. We express this relation by ab. The following notion of coincidence is more important.

(Heyting, Intuitionism: An Introduction, section 2.2.1.5, p.16)

Here the difference is that instead of every term “overlapping,” they are identical. But his definition of coinciding number generators in this other text is also similar, but it is technical and not entirely within my grasp:

The number-generators a ≡ {an} and b ≡ {bn} coincide, if for every k we can find n = n(k) such that |an+pb n+p| < 1/k for every p. This relation is denoted by a = b.

(Heyting, Intuitionism: An Introduction, section 2.2.1.6, p.16)

In other words, perhaps, although the terms of the two sequences may not be identically the same, if they are coincident, then they still converge upon the same value, and this is because, after a certain point, their corresponding nth terms will always fall within a gap smaller than any arbitrarily given one. For, it is saying |an+pb n+p| < 1/k for every p. In other words, his notion of the n terms “overlapping” may be made more mathematically precise, even though I am not exactly sure about its meaning. Or maybe what he is calling coincide here and “overlap” are equivalent to being identical and equaling, in the technical definitions. At any rate, we can say that one way or another, two number-generators coincide, and thus express the same value, when their series of terms are at least arbitrarily close if not equal. We will now look at his technical definition for the apartness relation.

‘For real number-generators a and b, a lies apart from b, ab, means that n and k can be found such that |an+pb n+p| > 1/k for every p.’

(Heyting, Intuitionism: An Introduction, section 2.2.3.1, p.19)

It seems to mean that two real number-generators are apart if after some nth term in their series, the succeeding corresponding terms will always be separated by some gap and thus each number-generator is converging upon a different value. And we symbolize the apartness relation between a and b as: ab. Let us look at some other definitions, moving to the simplest. This is from Heyting’s Les fondements des mathématiques, section 5.3.1.1:

a est différent de b, (a b), signifie, dans la terminologie de BROUWER, que a = b est impossible. Pour le continu, on a en outre la relation a est positivement différent de b ou a est | écarté de b (ab). Celle-ci est remplie quand, dans les suites d’intervalles qui définissent a et b, on connaît deux intervalles extérieurs l’un à l’autre.

(Heyting, Les fondements des mathématiques, section 5.3.1.1, p.24-25)

Here we define the apartness in terms of being “positively different” or being “apart from (écarté de).” a and b are apart when the series of intervals that define a and b, two external intervals can be found from one to the other. The series of intervals here seems to be the narrowing approximations we mentioned earlier. If there is an external interval or gap between a’s and b’s internal intervals, then they lie apart. Now we will give a positive definition of their equality. So again, rather than saying (or proving) that their inequality is impossible, we will consider a positive formulation.

Dans la théorie des nombres réels la relation ≠, étant négative, n’intervient pas. Il n’y a que la relation a = b et la relation de distance ab (voir ci-dessous “calcul numérique”). Le théorème “si ab est impossible, on a a = b” est remplacé par le suivant : “si a est distant de tout nombre c qui est distant de b, on a a = b”.

(Heyting, Les fondements des mathématiques, section 5.1.1.1, p.14)

So if number a is distant to all numbers c, which themselves are distant to b, then a equals b. For Griss’ formulations, we return to the other example in “Negationless Intuitionistic Mathematics, I” that in the above we set aside temporarily. Here he gives a formulation for different real numbers (in boldface):

Has the equation ax + by = 0 a solution for x and y, different from zero. i.e. a solution with at least x or y different from zero? The letters represent real numbers.

In intuitionistic mathematics they make a distinction between positively and negatively different with regard to real numbers. Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another; they differ negatively, if it is impossible that they are equal; you can only divide by a real number if it differs positively from zero. In negationless mathematics the idea negatively different is, of course, omitted. Therefore we mean henceforth by different positively different.

[...]

The result is:

ax + by = 0 has a solution different from zero, if at least one of the coefficients a and b differs from zero or if both are zero.

(Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

So again: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another.” Here, perhaps, we are saying that two real numbers are different if they have approximating intervals, perhaps something like parts of that triangular sort of shape of narrowing intervals in the Cauchy sequence, where one lies completely outside the other. So while two close real numbers may have many approximating intervals that overlap, at some point down the chain, there will be ones that do not overlap, that is to say, they lie completely apart from one another. Thus we can return to our current Heyting text:

Now the notion of different real numbers occurs in intuitionistic mathematics in two ways. In the first place it can be defined as meaning simply the negation of equality: two real numbers are unequal if it is impossible that they are equal; in the second place it can be defined in a positive way: two real numbers, defined by the number- generators a = {an} and b = {bn} are apart from each other (ab) if for some n, an and bn are separated intervals. Of course the second definition must be so understood, that we can actually find the number n. For Griss the first definition is useless, so he defines the relation of difference between real numbers as being that of apartness.

(94)

So here we see that two real numbers (two real number-generators) are apart from each other if there is some approximating interval that is separate from the corresponding one in the other series. Heyting then discusses a problem that I do not quite get. But I think it may be the following. Above, we defined inequality non-negatively, as being apartness. But, Heyting notes, there is then the danger of defining equality negationally as the impossibility of being apart. Instead, Heyting explains, Griss offered a positive formulation like we saw above, namely, two real numbers are equal if they are both apart from all the other real numbers:

One of the main properties of the apartness relation is: if it is impossible that ab, then a = b. This contains again the negation and hence must be replaced by a positive property. Griss found out that the following can take its place: if every real number c that is apart from a is also apart from b, then a = b. Let us call this property E.

(93-94)

Here is the full quote.]

The touchstone of a philosophical conception on the foundation of science is the actual development of the science in question on the basis of that conception; even the philosophical ideas which are involved gain in clearness and determination by their application to concrete problems. Too often philosophers content themselves with general ideas and leave the elaboration to specialists in the science; but in most cases the real difficulties occur in the application. Griss had the advantage to be at the same time a philosopher and a mathematician; after a short philosophical introduction he begins the construction of mathematics. Of course the main problem is to find a substitute for reasonings which involve negation; simply banishing these he would leave but insignificant ruins. In the first place the notion of difference must be examined, for each sort of mathematical entities separately. For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept. For rational numbers, which are defined as pairs of natural numbers, there is no more difficulty. For real numbers the case is different. A real number is defined by a convergent, contracting sequence of rational intervals; for the sake of brevity I shall call such a sequence a number-generator. Two number-generators a = {an} and b = {bn} coincide, if an and bn overlap for every n. Coinciding number-generators define the same real number; thus a real number may be defined as the class (in Brouwerian terminology the species) of number-generators which coincide with a given number-generator. All this is the same as in classical mathematics and has nothing to do with intuitionism or negation. Only Brouwer gave a larger interpretation of the word “sequence”; for reasons which I cannot explain here he admits that the members of a sequence are not determined | beforehand by some fixed law, but that they become determined one after the other, no matter how, for instance by free choices. Now the notion of different real numbers occurs in intuitionistic mathematics in two ways. In the first place it can be defined as meaning simply the negation of equality: two real numbers are unequal if it is impossible that they are equal; in the second place it can be defined in a positive way: two real numbers, defined by the number- generators a = {an} and b = {bn} are apart from each other (ab) if for some n, an and bn are separated intervals. Of course the second definition must be so understood, that we can actually find the number n. For Griss the first definition is useless, so he defines the relation of difference between real numbers as being that of apartness. But here a new difficulty arises. One of the main properties of the apartness relation is: if it is impossible that ab, then a = b. This contains again the negation and hence must be replaced by a positive property. Griss found out that the following can take its place: if every real number c that is apart from a is also apart from b, then a = b. Let us call this property E.

(93-94)

[contents]

Heyting, Arend. “G. F. C. Griss and His Negationless Intuitionistic Mathematics.” Synthese 9, no. 2 (1953-1955): 91–96.

.