The Analytic-Synthetic Dichotomy


Dragonfly

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Greg:

>Some time ago Daniel asked me to present my own summary of Peikoff's argument in the ASD.

Hi Greg,

I recall, perhaps incorrectly, that I was suggesting a formalisation of Peikoff's argument - as you were going to do it for something else less relevant - rather than a summary of it. But thank you for doing it anyway. I have read Peikoff's essay previously. It seems you have a formalisation in mind at some point. That would be useful if you get round to it.

> This should serve to remind us of the empirical origin of even our mathematical concepts.

Let's look at this issue from a slightly different angle: namely, origin from is, rather obviously, not the same as equivalent to.

To wit: even if it could be easily decided one way or the other, what does the genesis of the concept "triangle" matter? This is rather like saying there is no "dichotomy" between a bison and a jellyfish as we might speculate they all originated in some simple organism at the dawn of time. The cutting edge of the issue is, as you state generally (but then seem to take back in a rather ad hoc formulation which we will look at in a moment), that "perfect triangles" do exist in our heads but do not exist in the physical world. Thus there is a clear dichotomy, or at the very least a highly useful distinction. If all you want to argue is that there is some kind of original connection between the abstraction and the real world, well so does Plato. But this seems to me to be beside the point.

Incidentally, I do not get into debates over the meanings of words, but I would suggest a minor terminological clarification that might save confusion. When you write:

Greg: "...these beings do not exist in the physical world we should say that they don't exist at all, but we have thoughts of them..."

...it might be better to say these "beings" - such as perfect triangles - exist, but abstractly, and not physically.

Now, with that minor distinction made, what appears to me to be a rather ad hoc formulation is as follows:

Greg:"...there are surfaces that look perfectly triangular to the unaided eye, and from these we formed the concept of perfect triangles, without even needing to idealize, and then only later, when we measured carefully, did we find that the triangles were not composed of perfectly straight lines." (italics DB)

This seems simply confused, and merely verbalist, because when we "formed the concept of perfect triangles", we could have hardly done anything else but "idealised" them - obviously because the perfect triangle did not physically exist in the first place! Your theory becomes even weaker as you don't mention exactly what objects "look perfectly triangular" - in reality no object does, although there are objects that resemble perfect triangularity to a greater or lesser extent. However, to even make such a judgement seems to presume an abstract standard in the first place.

These are pretty straightforward criticisms of your position.

FYI, I am a Popperian, and find Popper's "3 World" cosmology provides a fruitful hypothesis for some of these and other perennial problems.

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Not if we take 'definition' is the broadest sense of the term, to apply to anything that indicates the meaning of a term. So a definition does not have to be verbal: there are ostensive definitions, which involve pointing.

When you think "point" what is it you are pointing at? When you think "line" what is it you are aligned with? A small spot of ink or graphite on a piece of paper is NOT a point. A stretched cotton or linen thread is NOT a line nor is an elongated streak of ink or graphite on a piece of paper.

There is literally nothing in the world some of whose properties you omit which will yield point or line. Which indicates that mathematicians (or anyone else) when they are -doing- mathematics are Closet Platonists. They can't help it. They are dealing with things nowhere in the physical world. Nowhere.

This, by the way, shows that Rand's notion of measurement omission is not quite on the mark. There is nothing in the world that has the property -length- such that if you omit its quantity you get the property -length-. What there -is- in the world are measuring sticks (rulers, yardsticks, tape-measures) and things in the world that we lay these devices against. And even then we are idealizing their rigidity. These devices are NOT rigid. They bend and stretch when sufficient force is applied, which is a state intermediate between nothing happening to them (a perfectly rigid object) or they break or become permanently deformed. Rand's error is a very plausible error and if I were to make a first guess at what length was, I too, would have said the same or similar thing. I am forced to use an operational definition of length. It is the average of several estimates of the difference between end marks on a measuring stick when I lay the stick against an object. Of if I go high tech, it is the difference between clock readings when I use a laser device to measure the time it takes for a light signal to bounce between one position and another or it is the difference in the time marks I record for two events. That is how "Mexican Radar" works. You time the interval it takes for a car to go between telephone poles at a known distance (see prior operational definition of length/distance) assuming that the car moves at near uniform speed (yet another abstraction that exists nowhere in the physical world).

In short our most effective tool for understanding the world, i.e. mathematics, is an exercise in applied hallucination. We are fantasizing. Which gets back to Wigner's question which he raised in his essay on the unreasonable effectiveness of mathematics. Rand never did give an answer to Wigner's question. Neither has anyone else.

Ba'al Chatzaf

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You cannot know that a statement is true unless it is either self-evident or provable by derivation (deductively or inductively) from the self-evident. Therefore you cannot know that an axiom is true unless it is self-evident or provable. Therefore you cannot know that it is an axiom unless it is self-evident or provable (note that I am not denying that it will still be axiom: statements can be true without us knowning them to be true).

I think you have axioms confused with postulates (see my last reply to Ba'al).

Really? Strange then that mathematicians for example talk about Peano's axioms and the axiom of choice. See for example here:

"In mathematics, an axiom is any starting assumption from which other statements are logically derived. It can be a sentence, a proposition, a statement or a rule that enable to construct a formal system. Unlike theorems, axioms cannot be derived by principles of deduction, nor are they demonstrable by formal proofs—simply because they are starting assumptions—there is nothing else they logically follow from (otherwise they would be called theorems). In many contexts, "axiom," "postulate," and "assumption" are used interchangeably.

As seen from definition, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results.

You may assume a postulate p to be true in order see its consequences, or even to prove it false. Then if you can validly deduce q from p you have proved the truth of the conditional statement 'If p then q'. But you will not have proven q, if you have not proven p.

You cannot prove or disprove a mathematical postulate (or axiom). Example: Euclid's 5th postulate.

Ttruths of physics and truths of mathematics differ in their subject matter, but not in their truth.

I know that the analytic-synthetic dichotomy is supposed to correspond to the mathematic-physics dichotomy, but I deny this--and do so even on your definition of 'analytic truth'--i.e., definitional truth--because I say that Newton's axioms can be derived from definitions of the terms 'body' and 'force', and so, even by your definition, are still analytic.

You may define ice in such a way that the statement "ice always floats on water" is an analytic truth. But that doesn't imply that it gives a correct description of the physical world.

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In short our most effective tool for understanding the world, i.e. mathematics, is an exercise in applied hallucination. We are fantasizing. Which gets back to Wigner's question which he raised in his essay on the unreasonable effectiveness of mathematics. Rand never did give an answer to Wigner's question. Neither has anyone else.

Ba'al Chatzaf

Korzybski proposed an answer to this question in Science and Sanity.

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In mathematics AND natural language 'point' would be considered undefined, and so one just 'knows' what it means.

How does know what they mean, without a definition?

Both could be defined, and in math terms normally are defined, and often need to be, because mathematicians often define terms differently from non-mathematicians (which is a very unfortunate habit they share with most other people: when people come up with a new concept they should come up with a new term for it--either a new word or a new combination of old words--but people are frequently lazy about that and grab a term that already has a meaning).

You cannot define every word in a given context, try it! Take any sentence and pick a word in it and define it. Then using the definition pick another word and define it. Keep doing this and you will find youself defining in circles and this means that at very basic levels of language we have to trust the person understands us. If we can't agree on some undefined terms then communication is not possible. In my mathematical definition of a circle above there is no need to define 'point', it understood what it means, on objective levels.

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In short our most effective tool for understanding the world, i.e. mathematics, is an exercise in applied hallucination. We are fantasizing. Which gets back to Wigner's question which he raised in his essay on the unreasonable effectiveness of mathematics. Rand never did give an answer to Wigner's question. Neither has anyone else.

Ba'al Chatzaf

Korzybski proposed an answer to this question in Science and Sanity.

But is not the right answer. The question is still open. The question is this: how can a system that consists of objects nowhere to found in the real world be so effective in helping us to understand the real world. If you have answer that works, you will become famous.

Ba'al Chatzaf

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But is not the right answer. The question is still open. The question is this: how can a system that consists of objects nowhere to found in the real world be so effective in helping us to understand the real world. If you have answer that works, you will become famous.

Ba'al Chatzaf

Ah, but I think it IS the right answer.

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Talking about "the unreasonable effectiveness of mathematics" sounds like talking about the unreasonable effectiveness of evolution.

--Brant

Evolution is a natural process. Mathematics is an artifact. It is made out of whole cloth by humans. The basic objects of the mathematical systems we use in physics have no physical existence whatsoever, yet they can be used and are in fact indispensable tools for understanding the physical world. How come?

The question is still open and no one has come up with the definitive answer.

Ba'al Chatzaf

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Ba'al:

>Evolution is a natural process. Mathematics is an artifact. It is made out of whole cloth by humans. The basic objects of the mathematical systems we use in physics have no physical existence whatsoever, yet they can be used and are in fact indispensable tools for understanding the physical world. How come? The question is still open and no one has come up with the definitive answer.

Yes, Brant's comparison is basically misguided. Mathematics itself is an artificial system that has evolved, through conjecture, trial and error. We regard it as useful because it can be applied to physical reality, albeit roughly, in the service of some human goal. (Where Brant's comparison might be better is if we said the theory of evolution was like a mathematical formula ie a human creation that explains certain phenomena succcessfully, although also could possibly be improved on). But usefulness aside, there seems to be more to it than that, and this is what is intriguing.

Mathematics seems to have certain features that relate to reality very powerfully; it has others that seem to have nothing to do with it. It also has certain features that at first seem to have nothing to do with reality, then - sometimes hundreds of years later - turn out to be profoundly related to it. Therefore, exactly what we are supposed to actually take in practice from Objectivism's handwaving condemnations about "breaches" between such abstract systems and physical reality seems to be a moot point. What does this high-sounding posturing actually forbid in real mathematical practice, other than the commonplace injunction to treat long and complex chains of abstact reasoning with a degree of caution? Hardly a very profound or original philosophic insight.

That's why I think the whole palaver over whether mathematics is "grounded" in reality is largely without merit. For even this formulation naturally implies that the artificial, abstract system of mathematics transcends physical reality anyway. It is, after all, merely "grounded" in it. It's better to simply ask whether mathematics can be applied to reality; the answer is plainly yes to some considerable degree.

The similarities and differences between mathematical systems and physical reality is what is interesting in the first place. For it is both mathematics' transcendence of physical reality, combined with its often startling and profound correlations with it, that has prompted Platonic speculations to this day.

As Ba'al suggests, it remains an open and highly interesting problem that debates over bland vagaries like whether such systems are "grounded in reality" contribute next to nothing to.

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As Ba'al suggests, it remains an open and highly interesting problem that debates over bland vagaries like whether such systems are "grounded in reality" contribute next to nothing to.

When you say it's 'open' does this imply that some day it will be 'closed'? Do you think that other theories, like QM will not be 'open' some day? It is not a mystery why mathematics is useful in modelling nature, it's because it is a language devoted to relations and structure, which is the only possible content of 'knowledge'. If you look at any field of study the only 'knowledge' in it will be some application of mathematics.

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GS:

>If you look at any field of study the only 'knowledge' in it will be some application of mathematics.

Errr...literature, aesthetics, ethics, politics??....Hey, even semantics...;-)

So what knowledge does one find in literature, aesthetics, ethics, politics?? There is an interesting concept in general semantics called 'orders of abstraction'. In this context a propositon cannot include itself as an argument, but must be a proposition (abstraction) of higher order. It is modelled after Russell's theory of types. So I can make a statement about any field of study but I exclude the current statement to avoid a 'vicious circle'. Another way of looking at it is that knowledge of the world is mathematical but 'knowledge of knowledge' is not necessarily so.

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Talking about "the unreasonable effectiveness of mathematics" sounds like talking about the unreasonable effectiveness of evolution.

--Brant

Evolution is a natural process. Mathematics is an artifact. It is made out of whole cloth by humans. The basic objects of the mathematical systems we use in physics have no physical existence whatsoever, yet they can be used and are in fact indispensable tools for understanding the physical world. How come?

The question is still open and no one has come up with the definitive answer.

Ba'al Chatzaf

Do mathematics *act* on the physical world? Only through human media. Aren't we just talking about concepts? A way of seeing and understanding that is more than simply visual--sensory?

--Brant

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Talking about "the unreasonable effectiveness of mathematics" sounds like talking about the unreasonable effectiveness of evolution.

--Brant

Evolution is a natural process. Mathematics is an artifact. It is made out of whole cloth by humans. The basic objects of the mathematical systems we use in physics have no physical existence whatsoever, yet they can be used and are in fact indispensable tools for understanding the physical world. How come?

The question is still open and no one has come up with the definitive answer.

Ba'al Chatzaf

Do mathematics *act* on the physical world? Only through human media. Aren't we just talking about concepts? A way of seeing and understanding that is more than simply visual--sensory?

--Brant

My very point. Math is strictly in our heads. If humans disappeared from the Cosmos there would be no more mathematics. Math, being a branial thing, regulates our actions. It has no direct effect on the physical world outside our skins. In that regard, mathematics is like language, a purely human artifact, but the very stuff of which our understanding is made.

Ba'al Chatzaf

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That's why I think the whole palaver over whether mathematics is "grounded" in reality is largely without merit. For even this formulation naturally implies that the artificial, abstract system of mathematics transcends physical reality anyway.

Daniel,

I have a semantics quibble here. Maybe you would like to find a different term than "transcend"? How can you transcend anything without starting at the same place, i.e., the "grounds"?

Michael

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In that regard, mathematics is like language, a purely human artifact, but the very stuff of which our understanding is made.

Ba'al Chatzaf

I don't see how there could be much doubt that mathematics is some kind of language. The question is, in it's pure form, not applied, what does it represent? It represents various exact relations and structure and so is available for any scientist to use if they find an application for it.

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Crows don't think so. Here's a chimp that doesn't, either.

What does that have to do with mathematics? It's just a demonstration that the chimp has a relatively good memory. Do you think that a parrot that can say "A is A" really understands what it's saying? (I mean a parrot with real feathers.)

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Dragonfly,

You don't like that one? Here is a conclusion different than yours on that very same chimp: More Chimp Math?

You might be interested in this, too: Monkey Math Machinery Is Like Humans'

I am trying to remember a report I saw on TV of a monkey or chimp performing actual low-level addition, subtraction, etc. I will try to find it.

Unfortunately, we might have to conclude that mathematics actually does have something to do with reality other than coincidence.

Michael

EDIT: Here are another couple of links for the interested:

It's a Math World for Animals

The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene

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You don't like that one? Here is a conclusion different than yours on that very same chimp: More Chimp Math?

Just good memory.

You might be interested in this, too: Monkey Math Machinery Is Like Humans'
The only reference to math is in the title, the experiment is about the speed of recognizing smaller or bigger things.

Not relevant. That the behavior of animals is optimized is the result of a combination of genetics and learning, but doesn't imply that the animal is solving differential equations in its head. That is a particular thing that humans can do to describe the behavior. The cat that can jump accurately on some high point doesn't know anything about paraboles or conservation of angular momentum, it just jumps.

That is unreadable in my browser, but I recognized the picture of "Clever Hans", the horse that allegedly could do arithmetic. Of course it couldn't really add numbers, it merely reacted to subtle clues from its owner.

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Dragonfly,

You certainly have dismissive opinions. (Leftovers from your Rand days?) Thank goodness science continues investigating despite the nay-sayers.

:)

The cat that can jump accurately on some high point doesn't know anything about paraboles or conservation of angular momentum, it just jumps.

How on earth do you know that "it just jumps"? I observe cats studying distance, target, etc., before they jump. They may not use numbers, but they sure as hell look and act like they use a primitive form of measurement.

Oops... I forgot. I am observing and using induction to understand... (maybe like a cat before it jumps?)...

:)

(Incidentally, ordinal measurement is a form of math, i.e., smaller and bigger.)

Michael

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Here is some very interesting work on the cognitive abilities of chimps by Sally Boysen of Ohio State University:

What are they thinking?

There are some videos that can be seen on that site, also.

Here is a press release about closing the research department due to lack of funding:

OHIO STATE TO CLOSE ITS PRIMATE CENTER, RETIRE ITS CHIMPANZEES

From the press release:

Over the past two decades, Ohio State studies with the chimps have netted remarkable discoveries. One project recognized the animals' ability to perform rudimentary addition and subtraction while another showed the animals' capacity for altruism, a trait long thought to be only human.

Still another project showed the animals' ability to link symbols and models to everyday tasks, while even more recent work suggested the animals may be capable of the simplest form of reading.

I would need to search more, but I am pretty sure the video I saw of a chimp solving math equations involved Sally Boysen. It showed a chimp in front of a computer monitor with a math problem and multiple answers. The chimp rubbed its wrist across the correct answer to different problems time after time.

Michael

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Talking about "the unreasonable effectiveness of mathematics" sounds like talking about the unreasonable effectiveness of evolution.

--Brant

Evolution is a natural process. Mathematics is an artifact. It is made out of whole cloth by humans. The basic objects of the mathematical systems we use in physics have no physical existence whatsoever, yet they can be used and are in fact indispensable tools for understanding the physical world. How come?

The question is still open and no one has come up with the definitive answer.

Ba'al Chatzaf

Well, according to our little discussion both the tools and the understandings are in our heads, not the referenced physical world. But take a shovel, also a tool. Absent the human agency it too is just a thing, not a tool. If Neanderthals had no language--a speculation I presume--they had no chance against our murderous ancestors.

--Brant

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