A discussion of "generalized numerals"

[j_j]

EDIT : I didn’t realize that “generalized numerals” was already in use for something that I don’t intend here. I thought the phrase was mine — not already taken.

What are numbers ? I suggest that we might want to understand them as blurry equivalence classes of generalized numerals. These numerals are roughly equivalent in terms of the role they play in the world. Numbers are “roles” played by a plurality of numerals, including the “generalized” numerals associated with that number.

But how should we make sense of ordinary numerals ? Two non-repeatable pencil scribbles, obviously different things in the world, are treated as “the same thing.” In grammar school, your handwritten arithmetic is judged primarily by whether it plays by the rules of arithmetic. Your handwriting is conspicuously different than Sally’s one desk over. But you turned in the “same” answers. We are trained to ignore everything but the “category” of the scribble. This “category” is the role, is the “number.” This category “doesn’t exist.” It is “ideal.” It is only there in the way that we treat differing scribbles in the same way. We perform the denial of difference.

Now for generalized numerals. It’s first worth noting that knots in ropes and notches on sticks have served as numerals. Perhaps I loan you seven cups of grain. I tie a knot in a rope for each cup. The rope won’t go bad like the grain. It’s easier to store. It “mirrors” the “quantity” of the cups I loan you.

Cantor’s theory of the infinite is based on this simple tactic — on the idea of a “bijection” between collections of things. Each cup of grain is paired with its own knot in the rope. Each knot likewise “re-presents” or “mirrors” a unique cup.

When you eventually pay me back, we get the rope and check that there is a cup of grain for each knot. We don’t even need names for numbers here, though it is convenient to invent sound- numerals like the speaking of the word “seven.”

So a jar of seven pickles is a generalized numeral for the number 7. This may sound weird, but what is counting ? If not the “categorization” of a quantity ?

Finally, patterns of flowing electrons are also generalized numerals. Those who have studied computer science might consider the 64 bit “representation” of an unsigned integer. And a pool containing x gallons of water ( probably a fraction, if we are talking about actual measurement ) is also…a generalized numeral.

There’s a story about Edison disdaining Tesla’s use of fancy calculus to calculate the volume of a strange shape. Edison just filled the strange shape with water, and then poured this water into a marked cylinder to determine its volume. Clever.

I suggest that generalized numerals are how physics is able to talk about the world. A measurement transforms an indeterminate “physical” situation into a generalized numeral. A model predicts a generalized numeral, the measurement of an otherwise non-numerical or pre-numerical empirical situation.

Does this make sense to anyone out there ?

Do you have a rival theory ?

What are your objections or concerns ?

[bongofury]

@j_j

What are numbers ? I suggest that we might want to understand them as blurry equivalence classes of generalized numerals.

Before reflecting on what you meant by generalising, I thought you were proposing pure syntax, like Goodman and Quine in Steps towards a constructive nominalism, for example? That is, fictionalism about numbers: understand arithmetic as a game with non-referring (‘meaningless’) numerals.

But if the generalising is a way of setting up a semantics for arithmetic by way of including suitable physical (or phenomenal) objects (collections or lengths) as tokens of the numeral type itself (or as tokens of a type in a corresponding system), then that reminds me of the logicist programme as a whole? And any reasonably enlightened approach to maths education? Understand arithmetic as talking about patterns in the world.

Also, less systematically, Languages of Art encourages recognition of reference by the exemplifying object to the numeral, e.g. by the pair to the numeral “2”.

[j_j]

@bongofury

Before reflecting on what you meant by generalising, I thought you were proposing pure syntax, like Goodman and Quine in Steps towards a constructive nominalism, for example?

I browsed that and liked it. Am I a nominalist ? Man, that’s tricky. Because I still think there is something “ideal” that we enact and even “find in the world.” Just taking the world as a system of things involves the “ideal” — the breaking of the qualitative field into enduring interpersonal objects. I’m definitely against mindstuff and pure internal meaning. Opposed to the usual mathematical platonism.

I like the Russian constructivism of Markov, and I agree with the intuitionist critique of the LEM applied to infinite sets. I almost sort of a finitist, but I don’t think it’s worth formalizing. It’s more a sense that computation isn’t free and isn’t in the mind of God. And hasn’t happened until it happens.

I almost tried to work in a “material” theory of computation. An algorithm is a “blurry equivalence class” of “concrete” or “material” processes, including the minimally material but still pencil-on-paper “theoretical version.” Or even just the spoken sketch, which is still minimally material in the sense of qualitative ( sonic ). For me the best way to read “material” is as “qualitative” — as opposed to this or that “ideal” or “signified” stuff.

[Frank]

@j_j

I suggest that generalized numerals are how physics is able to talk about the world. A measurement transforms an indeterminate “physical” situation into a generalized numeral. A model predicts a generalized numeral, the measurement of an otherwise non-numerical or pre-numerical empirical situation.

I think when you say “generalized” you mean the same thing as abstract. Whether the abstract objects are useful doesn’t change this abstract character. Were you trying to get away from that?

[j_j]

@bongofury

But if the generalising is a way of setting up a semantics for arithmetic by way of including suitable physical (or phenomenal) objects (collections or lengths) as tokens of the numeral type itself (or as tokens of a type in a corresponding system), then that reminds me of the logicist programme as a whole? And any reasonably enlightened approach to maths education? Understand arithmetic as talking about patterns in the world.

You are definitely hearing me correctly. A jar of pickles is a numeral. A line of 64 pennies, heads or tail varying, is a numeral. If we treat it that way, read it as a binary numeral.

Turing machines have to be “interpreted.” Like reading a clock or reading a calculator. We “enact” that interpretation. This is what is “ideal”, partially in the sense of normative.

I do think it’s a good approach to math education. You have probably picked up that I teach math, and I love doing it. Because it can be made fun, joyful. I taught Markov’s approach to the Turing machine to art students. It was great.

I never thought of logicism that way. I mostly thought of them as attempting to bypass numerical intuition, to “derive” it from something linguistic (formalized of course.) But I sort of agree with Brouwer and Poincare and Bishop. And Kronecker. “God” made the integers. We are more sure of the natural numbers than we are of their proposed “foundations.”

Of course by “God made 'em,” I mean they are intuitively very comfortable. In the last couple of years, I have also fallen in love with the lowly fractions. My students probably shake their fists at me in private.

Real numbers (irrational numbers) are “ways of messing with fractions.”

[j_j]

@Frank

I think when you say “generalized” you mean the same thing as abstract. Whether the abstract objects are useful doesn’t change this abstract character. Were you trying to get away from that?

OK, so the math lingo use of “generalize” ( or at least mine here ) is sort of abstraction but more like expansion. So a generalized numeral is the concept of numeral expanded to include jars of pickles, patterns of current in microchips, and so on. Expanded beyond marks on paper. Basically I’m trying to connect traditional numerals to their cousins in the world. It’s through generalized numerals ( counted or measured things ) that traditional numerals get their relevance and utility. I mean that’s my tentative claim here.

===

Rereading what you wrote, I should mention that ( for me ) there is something “ideal” in math that can’t be scrubbed off. But the “ideal” is not the “mental.” I’m still trying to pin down the blurry phenomenon. But I definitely don’t want to call it “mental.” It is out there in the world.

If you look at Hilbert’s formalism, he uses the marks ||| for three, etc. Even reducing numerals to a single mark (unary) requires the “ideal repeatability” of that single mark |. Because he is not talking about a particular stain of pencil graphite on a particular piece of paper. The most intense formalism, in a flight from the ideal, cannot escape this form of it.

[Frank]

@j_j

But the “ideal” is not the “mental.” I’m still trying to pin down the blurry phenomenon. But I definitely don’t want to call it “mental.” It is out there in the world.

A mental object is something you hold in your mind in the present. So you could hold the thought that 2 is a prime number. What you’re holding in your mind in this case is a proposition: that 2 is a prime number. I have a theory about what propositions actually are, btw. Not sure if you’re interested.

An abstract object is beyond any one mind, though. Attempts to reduce them to physical things is an interesting project.

[j_j]

@Frank

A mental object is something you hold in your mind in the present. So you could hold the thought that 2 is a prime number.

This is of course reasonable. So I’m the weirdo on this issue. I prefer to “empty” the subject and talk instead of the “eject.” Your “internal” monologue (and mine ) is a “qualitative” chain of “transcendent” phonemes. That others can’t overhear this “internal” monologue is an empirical belief. Sort of like the privacy of dreams. I don’t mean that people can possibly mindmeld with other people. That I can’t really parse.

I mean rather that maybe a device could transcribe your internal monologue reliably. Or the shaman can somehow ( who knows ?) can reliably summarize your dreams.

What is the “thought” that 2 is a prime number ? Is it the imagination of the vocalization of that sentence ?

I can definitely related to “running algorithms” in my head. Like I would walk in the woods and think of ciphers. So I am not denying at all the mathematical imagination. But what the hell is imagination ?

[j_j]

@Frank

I have a theory about what propositions actually are, btw. Not sure if you’re interested.

Fire away ! That sounds perfect for this conversation.

@Frank

An abstract object is beyond any one mind, though. Attempts to reduce them to physical things is an interesting project.

Yes ! And I’d say that things in the world are “ideal” in the sense of “going beyond any one mind.” I parse this in terms of “moments” of objects. But basically we “meet in objects.” Objects are “generalized signs.” Saussure’s sound-image can be generalized. It’s just the “identity” of the shared object. Different speakings of the same word are like different seeings of the same cat.

I should stress — just to be understood — that I personally don’t like the typical use of “physical.” It’s too loaded with dualist baggage, with the two-layer theory of image over substrate, or of mind painted on matter. I’m for radical pluralism.

For me “physical” vaguely means “what any of us can access with our sense organs.” But even a “physical” chair is “ideal.” It is not reducible to a moment, to this or that perception. Or measurement. Its boundaries are vague, etc. We don’t have the same beliefs about. But we enact our sense of sharing it, of having it on the stage of the world with us.

For me it’s better to look into the contrast or tension between the sensory-emotional in its slippery continuity and the relatively discrete “idea” of the thing – the “logical substance” of the thing we can all talk about.

[j_j]

Another approach. The mark “3” on a piece of paper is a thing in the world. But there are pencils arranged like | | | on the desk next to that paper that is marked with a “3.” Which is the numeral ?

Basically it’s easier to make a convention mark as needed than it is to carry around a box of pencils to line up. And even reading the pencils as a number is conventional (involving counting or pairing, both learned.)

[Frank]

@j_j

What is the “thought” that 2 is a prime number ? Is it the imagination of the vocalization of that sentence ?

Imagine a teacher points to a whiteboard that has the numeral 2 written on it. He says, “That is a prime number.” Analyzing this, we have

1. The utterance: the actual sounds the professor makes.

2. The sentence he uttered. A sentence is a pattern that follows the rules of a certain language. In this case, the sentence was: That is a prime number.

3. The proposition expressed by the utterance of the sentence is that 2 is a prime number. How do we know this? We discern it by the context.

As for how we discern his meaning, I’d fall back to Meno’s Paradox. Plato concludes from the paradox that we understand each other because of a shared ground he calls the World Soul. We each arise into life from the same thing, so it’s a kind of panpsychism?

So the proposition is the meaning of the uttered sentence. It’s the content. If the uttered sentence is a trolley, the proposition is what it’s carrying. We retain the use of the word proposition in spite of the glaring ontological issue because we need it.

This need is associated with the central column of communication: agreement. If you and I agree that 2 is a prime number, what is it we agree to? The utterance? The sentence? No, it’s the proposition.

The proposition is an idea that is central to communication and truth. Attempts to abandon it never get far. So what is it?

My theory is we picture our interaction with the world as if we’re having a conversation with it. If I’m looking for my glasses, I’m asking the world a question. I want the world to answer me. When I find my glasses, I hear the world’s answer. This is basically what propositions are: it’s coming from the expectation of being able to communicate with the world, or framing our interaction as if it’s a kind of communication.

So the world speaks to me, and when I understand correctly, I call it a true proposition. When I misunderstood, it’s a false proposition. So this is what truth basically is: alignment with the world. Communion.

This explains some of the conundrums associated with propositions, such as the idea of the unstated proposition, like the size of a certain rock on the dark side of the moon. When I say there is some true proposition regarding this, I mean that I expect that I could hear the world’s answer under the right conditions.

More later.

[j_j]

@Frank

As for how we discern his meaning, I’d fall back to Meno’s Paradox. Plato concludes from the paradox that we understand each other because of a shared ground he calls the World Soul. We each arise into life from the same thing, so it’s a kind of panpsychism?

OK. I love bringing Plato into this. Your “world soul” is what I mean ( sort of ) by “the forum.” We are “in language together.” But this means we are in “marks and noises” together. And of course in the more inclusive “systems of objects,” which includes telephone poles and pepperoni pizzas.

I’m not sure how to approach the panpsychism reference. Perhaps you can elaborate.

@Frank

So the proposition is the meaning of the uttered sentence. It’s the content. If the uttered sentence is a trolley, the proposition is what it’s carrying. We retain the use of the word proposition in spite of the glaring ontological issue because we need it.

OK. Yes. This is the reasonable and etymologically cemented metaphor of the carrier and carried. Trans-late. The content ( the “signified” ) is moved from this expression to some other expression.

My own preference is to invoke an “equivalence class” of expressions. These expressions play the same role in the world. So the proposition is not same content of a plurality of expressions. Instead it is the same-enough role that is played by a plurality of expressions. Each expression as mark or noise is “individually” significant. This significance varies in subtle ways, but we learn what to prioritize. Life is largely about ignoring differences.

I might try to support my own approach this way. For you to point at the proposition ( about the primality of 2 ), you have to pick a particular expression. “2 is prime” would work.

This is like picking from 1/2, 2/4, 4/8, 8/16,…

Some expressions are more convenient but “ideally” they are equivalent.

[j_j]

@Frank

The proposition is an idea that is central to communication and truth. Attempts to abandon it never get far. So what is it?

I agree that propositions rare central. Brandom calls them something like the minimal units that we can take responsibility for. The thesis, the claim, the assertion. Somehow marks and noises “represent” the arrangements of non-mark and non-noise objects in the world, but also those, because we are talking about talk right now.

@Frank

My theory is we picture our interaction with the world as if we’re having a conversation with it. If I’m looking for my glasses, I’m asking the world a question. I want the world to answer me. When I find my glasses, I hear the world’s answer. This is basically what propositions are: it’s coming from the expectation of being able to communicate with the world, or framing our interaction as if it’s a kind of communication.

So the world speaks to me, and when I understand correctly, I call it a true proposition. When I misunderstood, it’s a false proposition. So this is what truth basically is: alignment with the world. Communion.

I’m with you on this. And so is Bakhtin. We are addressed by the world. And we can question it, direct our curious attention, troubleshoot, resolve ambiguity.

Finding your glass makes sense as non-verbal. But I agree with Heidegger that the world is already “articulated.” Then language is “built on” on this articulation. Your glasses are a unified thing, that you can search for. You might debate whether they are “physical” or “mental” but they are already specifically “your glasses.”

I like the alignment metaphor. Is this imagination lined up with or congruent to perception ? How does the content or meaning of a vocalization fit in here ? Is it not weird and amazing that we humans are so good at “sending the situation” to other humans with noises that can be heard around corners ? We share the world through sound. We have to share only the relevant essence of the situation, because we can’t squeeze all the information of perception into a little chain of words.

[j_j]

@Frank

This explains some of the conundrums associated with propositions, such as the idea of the unstated proposition, like the size of a certain rock on the dark side of the moon. When I say there is some true proposition regarding this, I mean that I expect that I could hear the world’s answer under the right conditions.

I basically agree with you. I am reluctant to use “true” here for technical reasons. So I’d say that I expect my belief to be determined, given certain conditions. I could be certain — so I expect — that a defendant was guilty or not guilty, if I could somehow be an invisible witness following him around on the day of the murder.

Let’s say the world answers you and your belief becomes certain. Is it logically possible (imaginable, etc.) that world itself could be contradictory ? As in it answers somehow else in a way that is incompatible or contrary to the way it answered you ?

To me this would only make sense as an exception. If the world was frequently poly-faced, then we wouldn’t even believe that we share the world. But is some discordance thinkable ? I have to make my own beliefs cohere. So it should be difficult for me to think of a slightly poly-faced world. But we could even formalize it. The formalizer would be taking a fiction god’s pov, which might confuse the issue. But perhaps you see where I am going here ?

[Frank]

@j_j

OK. I love bringing Plato into this. Your “world soul” is what I mean ( sort of ) by “the forum.” We are “in language together.” But this means we are in “marks and noises” together. And of course in the more inclusive “systems of objects,” which includes telephone poles and pepperoni pizzas.

What specifically is this forum? A set? Or what?

@j_j

My own preference is to invoke an “equivalence class” of expressions.

I think the main benefit of analytical philosophy (AP) here is that it just presents you with what your stance requires you to give up, which in this case would be agreement as we generally understand the term, and therefore the whole point of communication.

I first got into AP because I was curious about how they would handle an issue related to identity. After a few introductions to it, I realized that there are some things AP just isn’t going to address. But in the meantime, I absorbed some of the AP aesthetic, which is dry and rigid. So the outline of communication I presented is a standard anatomical analysis, in that it’s just naming parts. It’s similar to the way we might describe gravity while entertaining multiple theories about what it is.

You can’t approach the science of gravity by starting out rejecting the notion that objects fall to the ground. That description is just the anatomy of experience regarding gravity. If the theory you settle on takes you to a place where you’re rejecting the whole idea of falling, that’s fine. But the original explanandum, that we see things falling, stands.

This is the AP approach. Just start by being clear about the content of experience wrt speech and communication.

So it’s in that spirit that I would explain why you can’t substitute “equivalence class” for proposition. There’s very solid logical argument for why that will require you to give up the concept of agreement. Is this off topic or interesting?

[Frank]

I could drift endlessly down any of the lanes you opened up.

[Corvus]

@j_j

What are numbers ?

Numbers are elements of our language denoting the quantity of objects, time of intervals and age of folks etc. In other words, numbers are words.

2 is also two. 5 is five. I am sure Chinese folks write differently for five or two in their own language.

2 itself makes no sense. 2 apples, 2 o’clock makes sense, but not much.

I bought 2 apples. I had a coffee at 2pm makes sense.
70 on its own makes no sense at all. Socrates died when he was 70, makes sense.

We don’t need the arabian numberings. We can get away with just words depicting the numbers like one two three four five etc. Hebrews only use their letters for numbers. They don’t use the Arabian numbering 1 2 3 4 etc.

[Rob]

Yep. They are our descriptions of concepts to help us logically understand relationships. Logic is itself fundamentally coherent. A single number or person or entity makes no logical sense APART from relationship with other particulars.

[j_j]

@Corvus

2 is also two. 5 is five. I am sure Chinese folks write differently for five or two in their own language.

I agree at the the level of causal common sense. But let me push back at the level of theory.

“2” is not “two.” As strings or inscriptions they are different. So the “is” a “metaphor” for the equivalence. These two different strings play the same role — usually.

@Corvus

2 itself makes no sense. 2 apples, 2 o’clock makes sense, but not much.

But why would most of us agree that 7 + 3 = 10 ? We might view the sign “2” as indicating an abstract situation. We basically ignore that we are talking about apples or years while we calculate. The result of the calculation is then plugged back into the concrete world.

@Corvus

We don’t need the arabian numberings. We can get away with just words depicting the numbers like one two three four five etc. Hebrews only use their letters for numbers. They don’t use the Arabian numbering 1 2 3 4 etc.

Right. There is nothing magical or sacred or necessary about a particular set of numerals. Fibonacci helped convince Europe to adopt our current set of 10 digits.

[j_j]

@Frank

I think the main benefit of analytical philosophy (AP) here is that it just presents you with what your stance requires you to give up, which in this case would be agreement as we generally understand the term, and therefore the whole point of communication.

I might say ( to risk a generalization ) that analytic philosophy can be logocentric or word-obsessed. For instance, you prioritize agreement. But I might prioritize the cooperation of primate bodies in the world. Of course the trading of marks and noises by these primates plays a crucial role here. A group of hunters devise and enact a plan. The “life” of the signs in that plan are in the enactment. The hunters finally cook and eat meat. The marks and noises “prove themselves” this way.

@Frank

I first got into AP because I was curious about how they would handle an issue related to identity. After a few introductions to it, I realized that there are some things AP just isn’t going to address. But in the meantime, I absorbed some of the AP aesthetic, which is dry and rigid. So the outline of communication I presented is a standard anatomical analysis, in that it’s just naming parts.

I appreciate the style of AP, and of course I love logical positivism, which some consider one of its sources. I like “dry” conceptuality ( I don’t want the sugar and salt of spirituality and politics, etc.)

I appreciate this naming of parts. But even this is creative and expresses an interpretation. Basically I’m saying that the “given” is blurry and sometimes controversial. I enact my understanding of the interpersonal given. I reason from claims that I assume the other will not ask me to justify. But I can be surprised.

@Frank

This is the AP approach. Just start by being clear about the content of experience wrt speech and communication.

To me this is just as much phenomenology’s mission. And it’s hard work ! For instance, you assume that expressions have “content.” This is a popular approach, but others have offered what I take to be potent criticisms of this approach. And this is no small issue. Communication may even be the most important and most difficult issue.

@Frank

So it’s in that spirit that I would explain why you can’t substitute “equivalence class” for proposition. There’s very solid logical argument for why that will require you to give up the concept of agreement. Is this off topic or interesting?

It’s not off topic at all. I mean to the degree that it’s up to me, I’m more about following the jazz where it goes.