3 Dec 2019

Deleuze (ED) “To Have Done with Judgment” / “Pour en finir avec le jugement,” entry directory

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Deleuze, entry Directory]

 

 

 

 

 

Entry Directory for

 

Gilles Deleuze

 

“Pour en finir avec le jugement”

“To Have Done with Judgment”

 

 

 

 

 

 

 

Deleuze, Gilles. “Pour en finir avec le jugement.” In Critique et clinique, 158–69. Paris: Minuit, 1993.

 

Deleuze, Gilles. “To Have Done with Judgment.” In Essays Critical and Clinical, translated by Daniel Smith and Michael Greco, 126–35. Minneapolis, Minn.: University of Minnesota, 1997.

 

 

 

 

5 Aug 2019

Priest (CBS) “Dialectic and Dialetheic,” collected brief summaries

 

by Corry Shores

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

 

[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[Graham Priest, entry directory]

[Priest, “Dialectic and Dialetheic”, entry directory]

 

 

 

 

Collected Brief Summaries for

 

Graham Priest

 

“Dialectic and Dialetheic”

 

 

Introduction:

Dialectics Requires Dialetheism

 

Priest will argue that Hegel’s and Marx’s dialectics were based on dialetheia, that is, on true contradiction.

 

 

1

Why It Is Necessary to Argue This

 

Many scholars argue that Marx’s and Hegel’s dialectics involve a non-logical notion of contradiction or that contradiction is conceptual and does not obtain in reality. Priest, however, will argue that the logical sense of contradiction is fundamental to their philosophies of dialectic.

 

 

 

2

The Argument Against this Interpretation

 

The main argument against reading Hegel and Marx as dialetheists is that it goes against the basic restriction of classical logic that you cannot have contradictions. But this restriction is based on an assumption and is thus not a necessary one.

 

 

 

3

Dialetheic Logic

 

Dialetheic logic is just like orthodox logic except that it allows for true contradictions, and when there are true contradictions, we cannot infer from them any other proposition we want.

 

 

 

4

Motion: An Illustration

 

One way we can illustrate how dialetheic logic can apply to dialectics is by accounting for motion in a Hegelian way. An object in motion is at a certain point at a certain instant, but since it is in motion, in that instant it is already leaving that point. Thus it is both true and false that the object is at that point in that instant.

 

 

 

5

The History of Hegel’s Dialectic

 

If we look at three of Hegel’s influences – Neo-Platonists, Kant, and Fichte – we see that Hegel borrowed self-contradictory ideas from each of them. Thus Hegel is a dialetheist, that is, he believes that true contradictions exist.

 

 

 

6

Contradiction in Hegel’s Dialectic

 

In Hegel’s dialectical movement, contradictory categories result from one another and are conjoined. It is in this ways that Hegel is a dialetheist [someone who thinks that there exist true contradictions].

 

 

 

7

Contradiction in Marx’s Dialectic

 

 

 

8

Identity in Difference

 

Hegel’s dialectic takes the form of identity in difference, formulable as (a=b)&(ab). This is a variation on the dialetheic formulation A&~A.

 

 

 

9

Dialectics and Epistemology

 

 

 

10

Conclusion

 

 

 

 

 

 

Priest, Graham. “Dialectic and Dialetheic.” Science & Society 53, no. 4 (1990): 388–415.

 

 

 

.

22 Jul 2019

Priest (CBS) Logic: A Very Short Introduction, collected brief summaries

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest’s, Logic: A Very Short Introduction, entry directory]

 

[The following collects the brief summaries for Priest’s book. The directory of entries without the summaries is found here:

http://piratesandrevolutionaries.blogspot.com/2015/07/entry-directory-priest-logic-very-short.html

]

 

Collected Brief Summaries for:

 

Graham Priest

 

Logic: A Very Short Introduction

 

 

Preface

 

Logic is an ancient discipline that was revolutionized in the 20th century with mathematical techniques and is currently very useful in information and computational sciences. This book will give a brief, broad, and non-technical overview.

 

 

Ch.1

Validity: What Follows from What?

 

“Logic is the study of what counts as a good reason for what, and why” (Priest, 1). An inference draws a conclusion from premisses (or from a premiss). It is valid if the conclusion follows from those premisses. It is deductively valid if it necessarily follows, that is, if no other conclusion could possibly follow, and it can be determined as such when “there is no situation in which all the premisses are true, but the conclusion is not.” An inductively valid inference is based on reasoning given in the premisses, yet other conclusions could also follow instead.

 


Ch.2

Truth Functions – Or Not?

 

Our intuitions about the validity of inferences are often correct, but sometimes they are misleading. One such case is the inference: q, ¬q / p, for example, “The Queen is rich,” “The Queen is not rich,” therefore “Pigs can fly”. Since the conclusion seems logically unrelated, we might erroneously think it is an invalid inference. By rendering these sentences into symbols and computing their truth values, we can see that there is no instance when the premisses are true and the conclusion not-true (false), and thus indeed it is valid. But since there is no situation where both the premisses can be true anyway, it is called vacuously valid. We also learn the truth tables for negation, disjunction, and conjunction, which are based on the truth conditions for these operations. If a sentence is true, then its negation is false, and vice versa. A disjunction is true only if at least one disjunct is true. And a conjunction is true only if both conjuncts are true. But conjunctions and disjunctions in English do not always map perfectly onto these truth tables.

 

 

 

Ch.3

Names and Quantifiers: Is Nothing Something?

 

When we speak of things, we might refer to some specific thing by name, like if we say, “Marcus came to the party”. In this case, what we are saying refers just to this one named person or thing. Or we might speak broadly and universally of all of a group of things, like if we said, “everyone came to the party”. In this case, what we say of the people or things applies to all of them. Or, we might refer to some thing, but without designating it specifically with a name, like when we say, “Someone came to the party”. Here we are saying something about a person or thing, but we are not specifying which one. When we want to speak of some thing or another, as in, “someone is happy,” we could use the existential quantifier and formulate this as, ∃x xH, meaning, there is some x such that x is happy. Or if we wanted to say, “Everyone is happy,” we could write ∀x xH, meaning, for all x, x is happy. Note that from just one quantified sentence an inference can be drawn. For example, if all people are happy, then there is some person who is happy. By using quantification, we can settle debates in mathematics and philosophy.

 

 

Ch.4

Descriptions and Existence: Did the Greeks Worship Zeus?

 

A definite description specifies a thing satisfying certain conditions, for example, “the man who first landed on the Moon”. Descriptions can be formulated symbolically by the use of variables that are predicated. The overall formulation takes the form ιxcx. Here, the ιx means, “the object x, such that…”, and the cx gives the conditions specifying the object. In our example we could write ιx(xM & xF) to mean, “the object x such that x is a man and x first landed on the Moon”. Furthermore, we may treat the whole description as something that can take predicates, and we can use Greek letters to stand for the whole description, thus possibly making the above formulation simply μ. This abbreviation will help us examine the validity of the Characterization Principle (CP), which is used in the Ontological Argument for God. We describe God as having a variety of properties that specify God, with the final one being “exists”: ιx(xP1 & … & xPn). The CP says that a thing characterized by certain properties in fact has those properties, and thus the whole described thing is predicated by the properties given in the description. Symbolically this involves substituting all cases of x in the description with that description itself. In this formulation we would get: ιx((xP1 & … & xPn)P1 & … & (xP1 & … & xPn)Pn), which in part says that the object that is omniscient etc., and exists, is in fact omniscient, etc., and does really exist. Using the Greek letters we can render the above substitution as: γP1 & … & γPn. But there is an important rule this argument breaks, namely that any predication to a non-existing entity is false. If there is a God, then the predication that God exists is true; but if there is no God in reality, then this predication is false. This means that for the argument to work, it must assume the truth of its conclusion at the outset, and is thus invalid. Yet there are cases where this rule does not apply, for example in instances of fictional entities like Greek gods whose properties can rightly be predicated to their description even though the thing described does not exist.

 

 

Ch.5

Self Reference: What is this Chapter About?

 

Paradoxical and otherwise problematic instances of self-reference lead us to suspect that we have more options than the following two: 1) a sentence can be just true, or 2) a sentence can be just false. Consider the “liar” sentence, ‘This sentence is false.’ If it is true, then it is false; but if it is false, then it is true. Either way, it’s truth-value will contradict what it says its truth-value is. So we have option 3) a sentence can be both true and false. Or consider the “liar cousin” sentence, ‘This sentence is true.’ Normally the terms in such a declarative sentence refer to things or situations by which we may determine the truth or falsity of the statement, that is to say, whether or not the indicated situation holds in reality or not. So if we say, “this chair is red,” we look to the indicated chair and its color, and we determine if the sentence is true or not. However, the terms in “this sentence is true” does not point us to such a determining situation, since we are only able to make two equally viable assumptions about its truth value, namely, that it is either true or that it is false; but, we have no way to make the determination one way or another, since it will always be consistent with what it says of itself under both assumptions. It would seem that we have no grounds that would allow us to determine whether it is true or false, and thus we have option 4) a sentence may be neither true nor false. The classical assumptions 1 and 2 lead us to conclude certain inferences are valid when our intuitions say otherwise. For example, “The Queen is rich,” “The Queen isn’t rich,” therefore, “Pigs can fly” (q, ¬q/p). Our intuitions tell us this seems invalid. But by just using assumptions 1 and 2, it is valid, since structurally speaking there is no situation where the premises are true and the conclusion is false. For, the premises can never all be true anyway. However, under the new assumptions, particularly that sentences can be both true and false, q, ¬q/p can be valid, if q is both true and false and p just false. For, q is at least true and ¬q is also at least true. However, our intuitions tell us that qp, ¬q/p is valid, but the new assumptions deem it invalid. Yet, perhaps it only seems intuitively valid if we forget that there are exceptional situations where sentences can be both true and false. There are other problems with the assumptions. When we assume that the liar cousin, “This sentence is true,” is neither true nor false, that means it cannot be true, but it says of itself that it is true. And while we might go along with saying that “This sentence is false” is both true and false, we might not feel the same way about “This sentence is not-true”. Here, we might conclude that it is both true and not-true (and not just true and false), which is a stronger contradiction that we may not want to accept.

 

 

Ch.6

Necessity and Possibility: What Will be Must be?

 

We can modify a statement of fact to indicate whether or not the referenced state of affairs is possibly the case or necessarily so. Modal logic allows us to deal with these modifications formally. Suppose “it will rain” is p. We write, “Possibly it will rain” as ⋄p, and we write “necessarily it will rain” as ◻p. Unlike truth-functional operators (like negation and conjunction), these modal operators do not alter the truth values of statements in a mechanically consistent way. To formally examine modally modified sentences, we think of there being other possible worlds about which we may make the same statements of fact, and these statements may be true or false depending on which alternate possible world it is in. In one possible world, it does rain tomorrow. But in another, it will not. We say something is possible when in at least one other world this state of affairs is false. However, no matter what possible world we conceive of, in all of them, if it rains, then fluid is falling. Such things which cannot be otherwise, when for example they are governed by fixed laws of physics, are considered necessarily true; for, in every other possible world they are true. We can diagram these possible world situations using boxes. In one box we give the statements of fact and their truth values for one situation or world (this world for example), and in other boxes we give the statements and their values for the other possible worlds. This helps us see which statements are necessarily true or false in one world and which are possibly so. This manner of formulation helps with certain debates, for example, it allows us to see that Aristotle’s argument for fatalism is fallacious. The argument makes us think that there is nothing we can do now to change the future, and also, that there is nothing in the past that we can regret or feel responsible for. The reasoning is as follows. If it is true that something will happen, then it will happen no matter what. But if it is false that something will happen, it will fail to happen no matter what. Either way, whatever happens occurs no matter what. By formulating this using modal logic, we see that it infers something incorrectly. There is a difference between the following two claims: 1) it is necessarily the case that if it is true that tomorrow I will get in an accident, then I will get in an accident, and 2) if it is true that if I will get in an accident, then I will necessarily get in an accident. If we just look at the semantic references, both formulations seem to have the same meaning. But on the level of their logical structure they are making different claims, and also structurally the second claim cannot be derived from the first, which is what is needed for the argument to hold. Aristotle’s fatalist argument would want you to believe that in every possible world you will get in an accident tomorrow, which is not so. It even acknowledges that the opposite could happen. However, there is a way to twist this fatalist argument a bit to remove that fallacy, and we may wonder whether or not this modification provides a valid argument for fatalism. We first say that there is nothing we can do now to change the past. This implies that states of affairs in the past are irrevocably true and statements about those situations are necessarily true. Now, suppose we do get in an accident tomorrow. This means it is true now if we say that we will. Suppose further that we said it yesterday also. We can say now that in the past it was true that we will get in an accident tomorrow. This means that it is irrevocably true that in the past we will get into an accident, and thus it is necessarily true that we will.

 

 

Ch.7

Conditionals: What’s in an If?

 

Conditionals are of the form, “if a then c,” or ac. The first term is the antecedent, and the second, the consequent. Conditionals are false only if the antecedent is true and the consequent false, and they are true for all other value assignments. But there are many difficulties regarding conditionals, and some of which call into question the universal applicability of these value-assignments. For example, according to the truth table for conditionals, when the antecedent is false, then the whole conditional is true, regardless of whether or not the consequent is true. This means that the following two conditionals should both be true: “If Italy is part of France, Rome is in France” and “If Italy is part of France, Beijing is in France”. But intuitively, the second one seems false. So conditionals are not truth-functional, since a lot depends on the meanings of the terms. In order to evaluate them, we can use possible worlds, like with modal operators: “the conditional ac is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” Since Rome is by definition in Italy, that means in no possible world would it not be in France, were Italy to be in France. So that is why the first sentence is true. However, since Beijing is by definition a city in China and not a city of Italy, then in some possible worlds Beijing will not be in France, were Italy to be in France. And that is why the second sentence is false. Another problem with conditionals has to do with ¬(ac), which has the same truth table as ac, and in fact is called the material conditional and is symbolized as ac. But although we might think that we can infer ac from ¬(ac), this is not in fact a valid inference, and we can show this using the possible worlds analysis. The important difference between ac and ¬(ac) is that ac involves the relevance of a to c, where there is no such relevance implied in ¬(a&¬c). For this reason we can think of situations where ac will be false but ¬(ac) will technically true, thereby invalidating the inference. There are other cases too of inferences using conditionals that seem valid, and yet there are troubling counter-examples that call their validity into question.

 

 

Ch.8

The Future and the Past: Is Time Real?

 

We can use tense logic to analyze the validity of inferences that are based on statements referring to different moments in time. We first think of a one-dimensional series of situations arranged in their proper chronological sequence. We then think of statements of fact. They may or may not be true for one temporalized situation or another. Suppose a statement h is true only for the temporally situated moment s0. This statement refers to an instantaneous state of affairs, like the moment the first bullet entered Czar Nicolas’ heart. It will be false for all situations coming before and after that temporalized situation, since the event did not happen at those other moments. However, at a succeeding moment in the future, we can say truly that the event happened in the past. And likewise for a preceding moment in the past, we can say it will be happening in the future. We use the modifier P for past (“it was the case that”) and F for future (“it will be the case that”). So in moment s1, Ph is true, and for moment s-1, Fh is true. We can further designate temporal relations by compounding the modifiers. PPh would apply h to a situation coming before some other situation that is already in the past. FPh would apply h then to some situation coming after some other situation that is already in the past. Now, P and F refer to some determinate situation in the past or future. We can instead refer to all future situations with the modifier G (“it is always Going to be the case that”) and all past ones with the modifier H (“it Has always been the case that”). We can also make a model  for this tense logic by arranging in sequence a number of s’s, placing s0 in the middle, and counting up and down the subscripts on both sides. This allows us to evaluate inferences based on tense modifiers. One example is McTaggart’s argument against the reality of time. If time is real, then the past and future are real, and thus they do not present logical contradictions. We then consider a sentence that is true just for the situation at one time-point. This means it did not happen in two temporally distinct time-points, and thus it did not happen both in the past and in the future: ¬(Ph&Fh). However, time flows, and so before it happened, it was in the future, and after it happened, it was in the past: Ph&Fh. The concepts of past and future present a contradiction, and thus time is unreal. One may object to the second formulation and say that it pretends that, for one situation that is located at one time point, the event can be both in the past and in the future. So to clarify the problem, we might then compound the modifiers and write ¬(PPh&FFh) to mean that the event did not happen at some determinate point coming before another in the past and at the same time happen at some determinate point coming after another in the future. Those following McTaggart’s reasoning can then say that still, because of the flow of time, PPh will be true and FFh was true, and thus, in contradiction with the prior, negated conjunction, PPh&FFh. But, by using the tense-logic model, we can display visually that the McTaggart argument is mistaken. There is never a singular temporalized situation where both terms in the past&future parings are true. Nonetheless, as this is a model that spatializes the flow of time, it might not be adequate for dealing with this argument about time’s non-spatial flow.

 

 

Ch.9

Identity and Change. Is Anything Ever the Same?

 

Over time, something’s properties might change. But it might either keep its identity or it might take on another one altogether. This presents a difficulty for philosophy and logic, especially since identity is a foundational concept in our thinking. We first distinguish objects and their properties, and we note that the properties may be variable while the objects remain constant. The ‘is’ of predication (x is red, or Rx) is different from the ‘is’ of identity (x is y, or x=y). However, Leibniz’s Law [of indiscernibles] uses properties to define identity. If two things share the same properties, then they are identical, and vice versa. This is a useful law in most applications, as for example when we use it for substituting terms in algebra. There are some other instances that at first seem to cast doubts on the applicability of the law, but these cases can be shown in the end to be mistaken for other reasons. However, there is one case that presents a big problem for the Law. We assume that identical things always were and always will be identical. When an amoeba A splits into amoebae B and C, then A has transformed into two other things in the sense of it having taken on new guises. This means that before the split, B and were identical to A and thus were identical to each other. However, after the split they are non-identical. This contradicts the assumption that things that are identical always are so.

 

 

Ch.10

Vagueness: How Do You Stop Sliding Down a Slippery Slope?

 

A thing can change gradually over time. A true statement about that thing’s status at the beginning can later be false at the end of the development. But in many cases, it is not clear when exactly during that development the status changes without ambiguity. “Jack is a child” is true when Jack is very young and not true when Jack is old; but, when precisely in his young adult years does it cease being entirely true and instead “Jack is an adult” becomes entirely true? This issue is related to sorites paradoxes. Consider that “Jack is a child” is true at the beginning, and “If Jack is a child at the beginning, then he is still a child one second later” also is probably also true. That means by modus ponens, “Jack is a child one second later” is true. Using this same sort of reasoning, we can then conclude that Jack is a child two seconds later, and so on, meaning that he never ceases being a child. (We reiterate the structure, taking the affirmed prior conclusion that Jack is still a child in the  succeeding second, and use it as a premise in an argument of the same structure, allowing us to conclude he is a child in yet the next succeeding second, and so on infinitely).  One solution to these issues is to use fuzzy truth values. We can say for example that when he is 3 years old, the statement “Jack is a child” has a full truth value of 1. At 9 years “Jack is  child” has a truth value of 0.75. At 14 years, 0.5. At 19 years, 0.25. And at 24 years, 0. And when we apply truth functional operators to statements with  values between 1 and 0, we can determine the different resulting fuzzy values. Also, we can say that an inference is valid when both the conclusion and the premises meet a certain minimum level of truth value, which is determined by the actual context to which the statements apply. What we find then is that the sorites paradox does not hold when we use this fuzzy system. [For, in order for the modus ponens inference to work in all steps, we will need the minimum value to be 0 (in order to accommodate the final transitional step), which is too low to be meaningful.] Also, fuzzy values do not clear up the situation entirely, because we have the same problem when we need to determine precisely at what point the values change from 1 to something less than 1.

 

 

 

 

 

 

 

 

 

Priest, Graham. Logic: A Very Short Introduction. 1st ed. Oxford: Oxford University, 2000.

16 Jun 2019

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

 

by Corry Shores

 

[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”

[youtube page]

 

(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

[The Laws of Addition]

 

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!

(Written by Wildberger, youtube page)

 

 

 

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

[The Laws of Addition]

 

[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]

 

 

 

 

 

 

 

 

[General Commentary:

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.]

 

 

 

 

Bibliography:

 

 

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

 

Youtube page for this video:

https://youtu.be/-96tlu_sShM

 

Course Youtube Playlist:

Math Foundations A (1-79)

https://www.youtube.com/playlist?list=PL5A714C94D40392AB

Math Foundations B (80-149)

https://www.youtube.com/playlist?list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

Math Foundations C (150 - )

https://www.youtube.com/playlist?list=PLIljB45xT85AYIeGfDQwHM8i6PQEDnnTI

 

Norman J. Wildberger, youtube channel:

[njwildberger]

Insights into Mathematics

https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ

 

 

 

 

12 Jun 2019

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

 

by Corry Shores

 

[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?”

[youtube page]

 

 

 

 

 

 

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!

(Written by Wildberger, youtube page)

 

 

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]

 

 

 

 

 

 

 

[General Commentary:

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).]

 

 

 

 

Bibliography:

 

 

Wildberger, Norman J. (2009). “What is a number?”.  Part 1 of the course series:  Math Foundations. Video.

 

Youtube page for this video:

https://youtu.be/91c5Ti6Ddio

 

Course Youtube Playlist:

Math Foundations A (1-79)

https://www.youtube.com/playlist?list=PL5A714C94D40392AB

Math Foundations B (80-149)

https://www.youtube.com/playlist?list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

Math Foundations C (150 - )

https://www.youtube.com/playlist?list=PLIljB45xT85AYIeGfDQwHM8i6PQEDnnTI

 

Norman J. Wildberger, youtube channel:

[njwildberger]

Insights into Mathematics

https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ