Philosophy 302a: Philosophy of Mathematics [under construction]
Please note that this is still largely an outline of the intended article.
"When you make the mistake of adding the date to the right side of the accounting statement, you must add it to the left side too" - Accountant's maxim
"Form follows function" - Architect's maxim
"Measure twice, cut once" - Carpenter's maxim.
"I went off to college planning to major in math or philosophy-- of course, both those ideas are really the same idea." - Frank Wilczek (1951 - )
Contents
Introduction
Part One: Mathematics the product
Section n: Sorts of maths
Section n: Analyses of the sorts of maths
Section n: Theories of the nature of maths
Part Two: Mathematics the process
Section n: Roles in maths
Section n: Values and disvalues
Section n: Means to maths
Section n: The number scales - a thought experiment
Section n: Basic quantitative concepts
Section n: Why maths will not reduce to logic
Section n: Maths compared with language
Section n: Maths contrasted with language
Introduction
In what follows, my use of G1, G2 and G3 refers to my metaphilosophical thesis that the central concepts in each of the subject matters of philosophy can be placed, non-arbitrarily, into three groups.
Part One: Mathematics the product
Section n: Sorts of maths
G1. Counting
G2. Structuring
G3. Measuring
Section n: Analyses of the sorts of maths
G1. Analysis of the valency or counting scale
G1.1: Scale definition: Unit/number. Unit value: U = 1. number value: n =) 0. That is, unit equals one divisible by n where n is any number (whole or fraction) other than zero. [My keybord lacks a 'does not equal' key.] For any value of n, this definition generates the positive and negative whole numbers and fractions.
G1.2: Relation of quotient to object: quotient internal to object (defines one object *as* one object)
G1.3: Status of positivity and negativity: absolute (the counting numbers are absolutely positive or absolutely negative).
Modes of counting:
G1.3.1: Unitary
G1.3.2: Fractional
G1.3.3: Infinitesimal
G2. Analysis of the ordinal or structuring scale
G2.1: Scale definiton: Unit/number. Unit value =)0. number value = 0. That is, any value other than zero, divided by zero. For any value of Unit, indefiniteness is generated.
This definition means that structuring quotients are not values but place-holders for values.
G2.2: Relation of quotient to object: abstract (structuring math is abstract in relation to its objects)
Since math concepts are the most abstract of our concepts, and the structuring math concepts are again abstract in relation to their objects, the structuring math concepts at the extreme of abstraction.
G2.3: Status of positivity and negativity: relative (structural positivity and negativity are relative to (i) their defined parameters, (ii) the values assigned to structural locations.
Modes of structuring:
G2.3.1: sequencing (locating relative to singularity)
G2.3.2: coordinating (locating relative to grouped plurality)
G2.3.3: eventuating (locating relative to totality)
Examples of terms expressing structural positivity - negativity:
G2.3.1: initial - final
G2.3.2: central - peripheral
G2.3.3: present - future
G3. Analysis of the cardinal or measuring scale
G3.1: Scale definition: Unit/number. Unit value: U = n. number value: n =)0. That is, any value except zero, divided by itself*.
*Unless 0/0 = 1. As far as I can tell, the consensus is that 0/0 = 0. However, consider that for any number, n, n/n = 1, and 0 is a number, so 0/0 = 1.
This definition generates the measuring scale of numbers.
G3.2: Relation of quotient to object: external (the measuring numbers are external to the objects they measure)
Measuring units are defined by an object or event of some sort.
G3.3: Status of positivity and negativity: relative (positivity and negativity of measurement are relative to the measured sizes of objects, which sizes are relative to the choice of mensural unit)
Modes of mensuration:
G3.3.1: qualitative
G3.3.2: spatial
G3.3.3: chronic
Examples of terms expressing mensural positivity - negativity:
G3.3.1: brighter - dimmer, heavier - lighter
G3.3.2: longer - shorter, wider - narrower, taller - shorter
G3.3.3: earlier - later
Note that most such terms have both positive and negative senses.
Section n: Theories of the nature of maths
G1. Maths as discovered (objective)
G2. Maths as conventional (intersubjective)
G3. Maths as invented (subjective)
Part Two: Mathematics the process
Section n: Roles in maths
G1. Observe and count
G2. Theorize and structure
G3. Move and measure
Section n: Values and disvalues in math
G1. Accuracy
G2. order
G3. Precision
Section n: Means to maths
G1. Analogy and accuracy
G2. Imagination, deduction and order
G3. Induction and precision
Section n: The Number Scales - A Thought Experiment
You can do the following as purely a thought experiment or as a practical experiment - I recommend the latter.
G1. The valency scale
Gather 30 dimes, sit down at your kitchen table, and put the coins in front of you in three piles of ten each. Think of these as three dollars that you have positively as wealth. Now visualize another three bucks next to them that you owe as debt. Now think of the three bucks in coins as the positive numbers, 1, 2, 3, and add an 'etc.'. Think of the three bucks you owe as the negative numbers, -1, -2, -3, etc.
The numbers on this scale are the counting numbers. We use them to count both positively and negatively. Any particular number on this scale is only positive or only negative, it cannot be both. And the number zero, being neither positive nor negative, is not on this scale.
Now, the math expression that defines a unit (a buck) on this scale is unit/n [that is, unit divisible by n], where unit = 1 and n is any number *except* zero. This definition allows us to count in both positive and negative units and positive and negative fractions. If we divide one of our bucks by ten we count a dime, and if by 100 we count a cent, etc.
With this scale we can count as many positive or negative wholes as we like, and as many positive or negative fractions as we like.
This scale is *absolutely* bi-valent. Any particular whole or fraction on it is absolutely positive or absolutely negative, it cannot be both positive from one point of view and negative from another point of view. There are no 'points of view' on this scale.
Next, wrap a dollar of dimes in duct tape and toss it into the sink. Go on.............. I'm waiting.
Okay, having done that makes it easier for you to grasp the idea that one counting unit defines one things *as* one thing. Counting those dimes as one dollar has defined them as one thing.
If you've thought this over, you're now ready to unwrap those dimes and go to the next stage of our experiment.
G2. The ordinal scale
Now, move your three piles of dimes back a foot or so and lay two or three pages of A4 paper end to end across the table in front of you. Next, take 15 of your dimes and line the 15 up across the table on the sheets of paper. You must line them up - this is important.
(Did you line up your three one-buck piles too, at first, even thought I didn't say to? If you did, it's understandable, but it wasn't necessary.)
Now you have a buck-fifty lined up in front of you. But the amount isn't important for the moment. What's important is to think of the dime on, say, your extreme left, as the first dime, the one next to it as the second dime, and so on.
The numbers on this scale are the ordering numbers: first, sixth, eleventh, third, etc.
Next, pretend that those dimes on the paper have circles drawn on the paper around them and you have taken them away again and just left the circles there. (It would be even better if you actually drew the circles on the paper, but I don't demand it.) Now you are ready to understand the math definition of a unit on this scale.
The definition is: unit/n, where unit = any value other than 0 and n = 0. Since any value divided by zero gives an indefinite outcome, the outcomes on this scale are not numbers, rather they are place-holders for numbers. The fifteen circles, real or imaginary, on your pieces of paper are place-holders for any amounts you might want to put on them.
Although there are no negative places on this scale, the scale can have positive and negative sections, depending upon the values placed in the place-holders and upon how you choose to define positive and negative.
Let's suppose you choose to define positive as '+v, where 59 < v > 91' (that is, any number larger than 59 and smaller than 91 is positive), and let's suppose that each dime in the line of dimes is worth 10p, where p is the place number it is in. By these definitions, the dimes in the sixth, seventh, eighth, and ninth places are positive dimes and the others on each side are negative. The dimes on the low side are negative because they are worth too little, and those on the high side are negative because they are worth too much.
G3. The cardinal scale
How wide is your table? Is it -263.195 wide? Of course not, it can't be 'minus wide'. Is it +16.75 wide? No? Well, you can't use numbers on the valency scale to measure it. And it isn't fourteenth, sixth, or eleventh wide - that too makes no sense. So you can't use numbers on the ordinal scale to measure it.
Now, start lining up your dimes one at a time up against each other in a straight line across your table, and we'll suppose, for the sake of the argument, that your table is exactly thirty dimes wide. You can see where this is going.
Numbers on the cardinal scale are the measuring numbers. And the value they have depends entirely upon what your unit of measurement is.
The unit definition on this scale is unit/n, where unit = n and n = any number *except* 0. And, since any number (except, perhaps, zero) divided by itself equals 1, outcomes on this scale are as big as you choose them to be. Your table might be thirty dimes wide, or just under two lengths of A4 paper wide, or 20.6 inches wide.
There are no negative measurements, only positive. But if your table (not to get too Freudian here) is only thirty dimes wide, it is probably smaller than most kitchen tables, and, in that sense, is negatively sized.
The fascinating thing about the three number scales is that other things, apparently quite different things, such as moral values, sorts of visual representation, and musical compositions, can be shown to have characteristics closely analogous to the characteristics of these scales.
Moral value concepts structured as the valency scale: right and wrong
Moral value concepts structured as the ordinal scale: virtue and vice
Moral value concepts structured as the cardinal scale: good and bad
As far as I know, this analogy between the number scales, moral values, sorts of representation and musical compositions, etc., is my discovery.
Okay, anyone who's been following my attempt to outline this article will have guessed that math is not my strength. I have made several attempts at defining the number scales, but I think I'm now pretty close to having my definitions say what I want them to say.
If any of you feel qualified to correct any remaining math errors, please offer me your corrections. I might or might not accept your correction, but I will be grateful for the offer.
Wow. My brain hurts. I think I sprained a lobe. I'll try to walk it off.
You may not be using "arbitrary" in a mathematical sense when you say "non-arbitrarily" but unless you are working from previously determined definitions then everything is arbitrary... right down to the system you choose to use.
G1.1 I think I understand what you mean when you are defining "scale" but what are you using to define scale in the first place? Integers? Rational numbers, Irrational numbers, Groups, geometric shapes, commemorative key chains? And if, for the unit value, "U = 1" then why not say the scale definition is "1/n"?
G1.2 What does "internal" mean?
I can't imagine counting infinitesimally. 1, 1 and a very little bit, 1 and two very little bits... Wait a minute! "Two" is too large. That should be "1 and a little bit and a little bit of a little bit."
G2.1 You cannot divide anything by zero. That's by definition. It's not like white with fish or after Labour Day. It is an immutable thingie. I understand napkin rings, bookmarks and seating at weddings (just barely.) I don't understand what a "place-holder" is. And if quotients aren't values but place-holders doesn't that mean that quotients have value (at least in the world of place-holders and napkin rings?)
Is "structural positivity - negativity" a binary distinction? If not then where does zero come in. I'm positive that I'm not full of negativity but then again not.
G3.1 * 0/0 is never equal to 0. You just can't divide by 0.
G3.2 If I knew what "internal" meant then I might have a chance with "external."
I had no idea that "mensuration" was a word. That is neat. On first reading I thought it might be a man's time of the month. No such luck.
Which reminds me of a mathematical question:
Q: How many women with PMS does it take to screw in a light bulb?
A: Would you shut your lousy mouth for once in your life!?
In the Roles in math section I think I know where you're going. But I can also imagine that moving and measuring comes before observing and counting. It's what made counting necessary. I am reminded of a Bill Cosby quote: "Lord, what's a cubit?"
Do induction and precision really go together?
As for Section n: you owe me a dollar. It's much cheaper than a plumber.
I know that there are good and bad cardinals. Richelieu would be bad. Albert Pujols would be good. Ozzie Smith would be great. Is the "cardinal scale" something along the lines of: "From 1 to 10, 1 being the lowest and 10 being the highest, how would you rate your experience with this survey so far?"
Is your discovery about numbers, morals, music et.al. an objective one? Or is it intersubjective and/or subjective?
Nice job. Smart (I think.) I'd rate it an '8' on a scale of 1 to 10, not including '9'. But I do think that a table can have negative width. Anyone who has put a tablecloth on upside down knows what I mean. If you don't believe me try putting your napkin rings on an upside down table. I'll bet you a roll of dimes you can't.
I admire all of the work you put into this. Thank you.
Thank you for these comments and questions - they are very valuable to me.
I started putting "non-arbitrarily" into all my articles on the Three Groups to emphasize my claim that each member of each trichotomy does have a defensible place in its trichotomy. My overall thesis is an attack on the ancient and deeply ingrained philosophical tradition ("fetish", as I call it) that each fundamental concept domain is completed by just two opposing or contrasting concepts. My thesis is that in most cases a domain is completed by not two but three concepts. I understand that in math there is more than one system of definitions, but I am not about a specific system, I am about what all specific systems have in common. Of course, I have to use a specific system (or invent my own) to try to express what I'm getting at.
I am using the rational integers in my definitions, and I am now aware that I have to remove "fractions" from those definitions. I use "Unit" instead of "1" because it doesn't equal 1 in each definition. Think of "Unit" as "Whole" rather than 1, except where I define it as 1.
What I'm trying convey by "internal" is that things countable as one thing carry that oneness with or 'in' them (hence the "throw the dimes in the sink" illustration - which obviously is insufficiently illustrative). This contrasts with the externality of measuring. A measured thing does not carry its magnitude with it. Its magnitude is instead in the thing by which it is measured - the measuring unit. For example, nothing is measured as having an absolute size; it's measured size depends on the unit of measurement with which we measure it. One thing can be measured using several different measuring units, each of which which gives it a different measurement. Contrastingly, a thing that is countable as one thing has that countability absolutely, by which I mean its countability defines it as a thing. It may in fact be a bundle of things (e.g. a bundle of dimes) but, likewise, each of those things (dimes) is defined as a *thing* by its own countability. Think of it this way: all counting, although we can count things in pairs, tens, dozens scores, hundreds, etc., is *ulimately* counting of things defined as single things - as 'units', if you will. When you count a thing, the count is always 'that's one' - when you measure a thing, the measure depends upon what you are measuring it against.
Counting infinitesimally. Isn't that, in effect, what calculus is about? Although one thing can be infinitely divided it remains one thing. It has limits. This is how the calculus is an answer to Zeno's paradoxes. Example, Achilles' race with the tortoise. Although the tortoise's head-start can be infinitely divided in theory, it remains a finite unit of distance, and as such it is, as experience invariably demonstrates, actually crossable by Achilles. Don't I mean "measuring infinitesimally"? No. The calculus counts, it calculates; hence its name. Measuring is not calculating - though, of course, once measuring is done, calculation can then be applied to the measurements.
This starting to strain my brain too. I'll have to take a rest and get back to it later.