I have set up this page so that the extensive comments on foundational questions, which have been appended to my post on the commutative law, can be put here, where they will be more easily accessible.
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Peter Cameron's Blog 
[My thanks to Professor Cameron for creating this separate page]
I thank Mr. Ralph Dratman for his response.
Mr. Dratman writes: “To Ross Templeman, I am not using a pseudonym. My real name is Ralph Dratman. Why do you write that my comment was a “wind up” or that I have lost my marbles?”
My reason for assuming that a pseudonym was being used is that a lower case single word has been used as a designation and would not be a first name I am familiar with even if a capital letter had been used. If I used templeman as my ‘screen name’ then that would still be a pseudonym to my mind.
I used to spend some of my spare time posting on various scientific pages on websites and have come across more trolls than one could shake a stick at. More often than not their posts sound a lot like Mr. Dratman’s initial contribution (although there occasionally seemed to be genuine conspiracy theorists etc.) and it is typical of trolls to use pseudonyms. As it happens the marbles remark wasn’t meant seriously, but it doesn’t seem to have come off that way, so I withdraw it.
To be fair however I am not sure why anyone else would interpret Mr. Dratman’s original post as being anything other than a confusing denial that basic arithmetic is true. This is a claim far more radical than sweeping rejections of Evolution, or a claim that we never went to the moon, or a claim that geology is nonsense and Noah’s Ark is clearly the truth, or that the government is allowing aliens to abduct citizens for medical experimentation.
Mr. Dratman’s latest post, when combined with his original contribution, clarifies his position; he is a radical empiricist, perhaps sympathetic to such thinkers as John Stuart Mill and Willard Quine. A view which commits him to believing that alleged abstract truths are nothing more than conventions and that the only credible source of information is observed regularities in nature. I believe that radical empiricism fundamentally misunderstands the status and nature of mathematics.
I think the best response to this sort of thing is to try and clarify an alternative that is more akin to common sense.
Let’s consider what mathematics is. I think it is easiest to approach this issue in an iterative fashion, by which I mean we will give an initial (very) rough definition, reflect on it a little and then provide a better one, followed by a repetition of this process until we have something that is satisfactory for our purposes. [I will not concern myself with being as thorough as a modern mathematician might demand however and I will be informal in my use of the word ‘relation’].
Let us begin with the following VERY rough definition; ‘mathematics is the study of relations’. ‘What is a “relation”?’ one might well ask, and what does it mean to “study” them?
Well, let us start by trying to give examples of “relations”:
1) On the table in front of me are six coffee mugs arranged in a hexagonal pattern.
2) Some of the mugs are of the same colour and some are not.
3) Four of the mugs contain different known quantities of liquid. One contains an unknown number of pieces of chocolate and the other contains ‘nothing’.
4) Five of the mugs are made of ceramic and have a disposition to being broken if they are knocked off of the table whereas the other is made of metal and has no such disposition.
5) A seventh mug then appears in the center of the hexagonal arrangement, but upon further inspection it is revealed to be a hologram and so is not real, whereas the original six are real.
That’s enough for the moment I think. There are quite a few relations in there, some more implicit than others. However a problem with our initial definition is apparent, namely that not all of these relations are of interest to mathematicians. For example, ‘a disposition to being broken’, is not a mathematical concept as far as I am aware. Nor is ‘being real or illusionary’, or ‘being made of ceramic or metal’. However ‘hexagonal pattern’ is, as is ‘six’ and the notion of ‘some’. An ambiguous case is the notion of being ‘known or unknown’, which is certainly something that comes up in mathematics all the time, but cannot be claimed as unique to the field.
So what distinguishes the relations that are of interest to mathematicians from those that are not? The famous Swiss mathematician Leonhard Euler would have said that mathematicians are concerned with the study of *quantity* by which he meant; relations capable of being increased or diminished.
Let’s use this as an improved (but still too rough) definition; mathematics is the study of relations that are capable of increase and/or diminishment. We still haven’t addressed what ‘study’ actually means, so let’s make an initial remark about that.
Well let’s use chemistry, the study of relations concerning substances, as an analogy. One of the tasks of a chemist is to assign different substances to categories based on the sharing of a common characteristic(s) (acids and alkalis for example), where the word ‘characteristic’ is used in a very broad sense. Another is to observe and analyse how different substances behave under prescribed conditions. Another is to determine the underlying structure of substances.
Various relations are implicit in such endeavors, such as:
1) Whether or not a chemical can be decomposed into simpler ones.
2) The number of electrons possessed by particular elements.
3) The variety of possible chemical bonds that different elements might form.
4) The comparative rates at which chemical reactions occur.
5) The comparative ratios of different elements composing a substance.
6) The search for ‘new’ substances.
Analogous relations occur in arithmetic, for example:
1) Whether or not a particular whole number is equal to a sum of squares.
2) The number of prime factors that a given whole number ‘possesses’
3) The variety of resultant numbers that a finite collection of whole numbers can ‘produce’ using just addition and subtraction.
4) Given a sequence of numbers that increase in size according to a rule, what measurements can we provide regarding how the numbers increase in size ‘on average’?
5) Relative proportions of odd and even primes that are factors of a whole number.
6) The search for ‘new’ kinds of whole number.
So we can provide a more detailed (but still rough) definition of mathematics as follows; Mathematics is the discipline that seeks to categorize relations that are capable of increase and/or diminishment, seeks to analyse how such relations behave under prescribed conditions and seeks to understand complex relations in terms of simpler ones.
So there are certainly similarities between chemistry and mathematics here and presumably such similarities provide some of the motivation for a radical empiricist stance such as Mr. Dratman’s. But there are important differences as well.
For one thing there is a kind of asymmetry between something like the notion of number and the definitions and assertions that we find in scientific theories such as chemistry.
Protons were once conjectured to exist but were then shown to be three ‘quarks’ bound together by the ‘strong nuclear force’. But if Protons don’t actually exist, what conception were scientists carrying around in their heads all those years? Were they trying to imagine some kind of absurdity such as the ‘round-square’? Clearly this is not the case.
The only feasible explanation for this kind of phenomenon that I have ever encountered, is that our mind is ‘equipped’ with certain concepts that we can use in conjunction with the ‘data’ of our five senses, in order to devise ideas. In the case of something like the proton, we might use the general notion of an ‘individual thing’ for example. Other examples might be concepts like; ‘two’, ‘possibility’, ‘change’, ‘identity’, ‘extension’ and various others. What they have in common is that they are all plausibly thought of as kinds of relation.
Concepts such as these are what we use to construct many kinds of *definitions*, whether those definitions are concerning the physical sciences or mathematics. Every definition is subject to the constraint that it does not entail a self-contradiction and that it is meaningful, so ‘round square’ does not define anything for example and neither does ‘green hope’ . When we combine definitions with further assertions, we produce theories.
This leads to the question of how we know that a definition will never entail self-contradiction. The general question is too difficult for me, but since Mr. Dratman has specifically questioned the commutativity of the natural numbers let us restrict our attention to this sort of thing. That is to say we ask how definitions constructed from our given intuition of ‘number’ and some others, can be shown to not entail contradiction?
Well the answer is implicit in Professor Cameron’s blog entry regarding the commutative law; we know that such definitions do not entail contradiction through the use of the Principle of Mathematical Induction. Since this post is already very long I won’t repeat the details of that here.
Now a position such as that held by Mr.Dratman must assert that the Principle of Mathematical Induction is a disguised empirical theory. The easiest way to reply to this, I think, is to show that it is founded on intuition in a way that is sufficiently simple for the soundness of the principle to be clear. Well, what is it about the principle that might be unclear?
The principle concerns a kind of relation between propositions, specifically an unending number of them. Mr. Dratman’s contributions suggest that it is the ‘unending’ part that bothers him, that is to say he is a type of finitist. Well I think that problem is easily disposed of using an example like the ‘Russian doll rectangles’ I used in a previous post. That is to say our intuitions of a continuum are sufficient to dispose of the difficulty.
Let us now address the specific points made by Mr. Dratman. Mr. Dratman writes:
“As we all know, results calculated in Euclid’s geometry turn out not to agree numerically with large-scale measurements made in our actual world.”
According to the view of mathematics I have (very crudely) outlined, the validity of General Relativity does not falsify Euclidean Geometry, it falsifies the theory that the principles of Euclidean Geometry always apply to physical space. Euclidean Geometry itself only requires that we have an intuition of a continuum and can construct definitions of the kind that Euclidean Geometry uses.
The only constraint on such definitions are that they are meaningful and do not entail self-contradiction. That they do not entail self-contradiction was established by David Hilbert. His demonstration relies on our intuitions of number and the Principle of Mathematical Induction.
Mr. Dratman later writes:
“Let me put that another way. The simultaneous existence and non-existence of specific intermediate particles must sometimes be taken into account to make predictions about even a single event which both began and ended with one particle.”
I am afraid I cannot see the relevance of this sort of thing to the issue of the soundness of arithmetic. The process begins with one ‘actual particle’; this is then followed by a superposition of two possible future states of the universe and then ends with one actual particle again. So we are switching between modal and categories mid-stream as well as changing contextual definitions (particle on one hand, events on the other). I cannot see how this might cast doubt on arithmetic.
I think that will do for now. A more thorough response might be desirable, but this is not my blog and I think this post is long enough as it is.
Regards.
My point was not that GR invalidates Euclid. No scientific theory or experimental observation can invalidate a mathematical system. I was pointing out that our observations of natural events cannot be assumed to match a preselected mathematical theory. Rather, the mathematical models used by physical theories are constructed expressly to fit experimental results as closely as possible.
Einstein labored to find a geometric framework whose parameters could be set to describe not only the local behavior of space and time near a particular event, but also the metric connections among the neighborhoods in a larger region. The result of his heroic effort is known as General Relativity. It is an extension and generalization of Euclid’s geometry, not an invalidation thereof.
About commutativity, I was (clumsily) trying to point out that experimental counting of dots on a grid can neither affirm nor falsify the status of commutativity within mathematics. Even if physical commutativity were found to fail experimentally on some large scale, say above 10^23 items, that scientific result could have no bearing on the validity of a mathematical system of postulates and theorems such as ZF.
Ralph Dratman
I have just noticed a typo in my post. Near the bottom it says “modal and categories” , it should just say “modal categories”
I thank Mr Dratman for his response.
Naturally I agree with Mr. Dratman’s latest statements regarding geometry. However I believe that they are in tension with some of the previous statements regarding arithmetic.
My reason for inferring that Mr. Dratman was a radical empiricist is the remarks regarding Quantum Mechanics. I have just reread all of Mr. Dratman’s posts on the commutative law thread to make sure I did not mistakenly misread something.
If I am to interpret them as advocating a more commonsensical position than I originally interpreted, then I must confess to being at a loss as to what the intended implication of the of the Quantum Physics example is. On this occasion I will not hazard to make my own guess. 🙂
Kind Regards
I guess what I meant to say was that we cannot know in advance where either experimental science or logical reasoning might lead us in the future — especially the far future. Today many millions of us own devices that could only be described as magical to humans just a few hundred years ago. How can we know what is to come?
Einstein said he believed thermodynamics was the only part of science that would “never be overthrown.” Very well, I put my chips on thermodynamics (and therefore the ultimate heat death of the universe), but I am not willing to bet on anything else.
I apologize if this disturbs someone, particularly thinking about Ross Templeton. What I wrote above might even have disturbed me, if someone else had written it!
I simply don’t believe I know anything in advance — not definitely.
I thank Mr. Dratman for his response (or at least I think the post was intended as a response to me). In what follows I will assume that pyrrhonian skepticism and close variants are false without argument.
Mr. Dratman writes:
“I guess what I meant to say was that we cannot know in advance where either experimental science or logical reasoning might lead us in the future — especially the far future. Today many millions of us own devices that could only be described as magical to humans just a few hundred years ago. How can we know what is to come?”
When it comes to experimental science I would counter that it very much depends on what issue one is considering. Is there any possibility at all that 45th century science will overthrow our current view that the Earth is not flat?
Mr. Dratman later writes:
“Einstein said he believed thermodynamics was the only part of science that would “never be overthrown.” Very well, I put my chips on thermodynamics (and therefore the ultimate heat death of the universe), but I am not willing to bet on anything else.
I apologize if this disturbs someone, particularly thinking about Ross Templeton. What I wrote above might even have disturbed me, if someone else had written it!”
Let it be known that Mr. Dratman’s positive commitment to thermodynamics does not disturb me in the least.
Mr. Dratman then continues:
“I simply don’t believe I know anything in advance — not definitely.”
I am not quite sure what Mr. Dratman is getting at with this sentence. Is he merely saying that he personally adopts the healthy attitude of (non-pathological) skepticism when it comes to certain issues or is something stronger being said?
Kind Regards.