The Analytic-Synthetic Dichotomy

[Michael Stuart Kelly]

Dragonfly,

You could say all that about concepts or any other abstraction. (But you don't "throw away" the starting point. You merely abstract it into a mental unit.)

How about the following for a specific axiom?

Abstraction exists.

Michael

[Dragonfly]

How about the following for a specific axiom?

Abstraction exists.

Does that imply that the subject of the abstraction exists?

[Michael Stuart Kelly]

Does that imply that the subject of the abstraction exists?

Dragonfly,

Sure. Here's a corollary:

Fingers and toes exist.

Michael

[Dragonfly]

And a 12345-dimensional sphere exists?

[Michael Stuart Kelly]

Dragonfly,

Does it have toes and fingers?

Michael

[Dragonfly]

No, so what is your answer?

[Michael Stuart Kelly]

And a 12345-dimensional sphere exists?

Dragonfly,

I have a question. Is it possible to conceive of "a 12345-dimensional sphere" without learning how to count something physical at the beginning? Do you know of anyone who can do this or has had this kind of experience?

I learned that the Objectivist theory of concepts considers knowledge to be contextual and hierarchical. Do you have a different explanation for this aspect of it?

Conclusion: If you find that in order to learn math, a person needs to learn how to count first, and probably that was fingers and toes, there you have it: a 12345-dimensional sphere with fingers and toes in the background.

Michael

[Dragonfly]

So you think that a characteristic of astrology must be a characteristic of astronomy, while the former is the "background" of the latter?

Does a contradiction exist? Does nothing exist?

[Michael Stuart Kelly]

Dragonfly,

You equate learning to count from fingers and toes with astrology?

Isn't that like cutting off your nose to spite your face?

Michael

[Dragonfly]

You have an incredible talent for focusing on the inessential in an argument and missing or evading the essential.

[Michael Stuart Kelly]

Dragonfly,

Since you have been unable to clearly explain such "essential," you are now going for ad hominem?

I personally am waiting to examine the example of a person who can perform higher mathematics without first learning how to count. I find counting (and I mean literally counting things) to be really, really, really essential to learning mathematics.

If you disagree, I am all ears. What is the "essential" I am missing?

Michael

[BaalChatzaf]

And a 12345-dimensional sphere exists?

Mathematically. I don't know of any concrete application (off hand) in which a 12345 dimensional sphere would model any constructable entity (abstract or concrete). On the other hand one could not say such an application is impossible.

Ba'al Chatzaf

[Brendan Hutching]

It will clarify things in this debate to ask whether you are talking about overall similarities between things or similarities in a respect (i.e., in respect to a given attribute). The latter is what is mainly important, but modern philosophers tend to think only of the former.

If by “overall similarities” you mean a general likeness or resemblance, I’ve been talking about the latter, ie ‘similarity’ in terms of an attribute. But as I understand it, overall similarities between things also have a bearing on this problem, because if one rejects real universals surely the next step is to discover some other reason for why we group some things and not others.

I’m not sure about your second comment. I note your comments about trope nominalism in another post, but I would have thought that conceptualists and other types of nominalists would be interested in ways of grouping things in terms of some type of general resemblance. Am I missing something here?

Brendan

[tjohnson]

Dragonfly,

Don't we have to learn how to count before we learn higher math? And don't we start by counting things we observe, like fingers and toes?

That's one connection to reality at the start.

Michael

Technically, when you use numbers to count things it is called applied math, albeit it's a very simple application. In pure mathematics there are no 'things' to count. One of the reasons mathematics is so unpopular in our education system is the belief that 'pure' math is not good for anything and we have to always apply it somewhere. Korzybski helped me understand that mathematics is important for far more than being applicable to some physical theory - he suggested that it was actually similar in structure to our nervous system. When you think about it, all languages represent something right? So what does 'pure' mathematics represent? It may just be possible that it represents the actual workings of the nervous system as it abstracts structure from our environment - quite a radical idea!

For example, integral calculus is a process that takes a bunch of 'static pictures' and integrates them into a continuously changing image. This sounds remarkably like what our eyes do with discrete nervous impulses from the stimulation by light. Some of the terminology of advanced mathematics is extremely suggestive - check out topology, category theory, fiber bundles, etc. and you might be surprised at what goes on in 'advanced' mathematics.

[BaalChatzaf]

Dragonfly,

Since you have been unable to clearly explain such "essential," you are now going for ad hominem?

I personally am waiting to examine the example of a person who can perform higher mathematics without first learning how to count. I find counting (and I mean literally counting things) to be really, really, really essential to learning mathematics.

If you disagree, I am all ears. What is the "essential" I am missing?

Michael

You are not missing a thing. Everyone's mathematical career begins with counts and shapes.

It is no accident that the earliest manifestation of mathematics pertained to keeping track of the seasons, counting objects and measuring land. The word -geometry- itself is Greek for earth-measure. Mathematics also emerges from symmetries. Early artists and designers worked out the seventeen planar frieze symmetries empirically. Spatial symmetry is an abstraction on shape and is independent of size (i.e. area and volume). This is surely an early instance of measurement omission.

Mathematics became "big time" when it was applied to motion. (Orseme, Descartes, Galileo, Kepler, Leibniz and Newton). That is the origin of modern physics. The mathematics is absolutely essential here. Consider that Aristotle had no precise way of expressing the idea of acceleration. That is because the mathematics for doing so did not exist in his lifetime.

Ba'al Chatzaf

[Michael Stuart Kelly]

Technically, when you use numbers to count things it is called applied math, albeit it's a very simple application. In pure mathematics there are no 'things' to count.

Tom,

Do you think it is possible to learn pure mathematics without first learning this simple applied mathematics? Do you know of anyone who has?

If not, in pure mathematics reality ("things") has been merely abstracted into symbols and the symbols are manipulated and structured. But the base of those symbols is reality, i.e., "things." Math is not divorced from reality. If it were, it would not work with reality.

I imagine that it is possible to make fantasy math with actual numbers just as it is possible to write about fantasy worlds with words meaning actual concepts.

Michael

[tjohnson]

Do you think it is possible to learn pure mathematics without first learning this simple applied mathematics? Do you know of anyone who has?

If not, in pure mathematics reality ("things") has been merely abstracted into symbols and the symbols are manipulated and structured. But the base of those symbols is reality, i.e., "things." Math is not divorced from reality. If it were, it would not work with reality.

I imagine that it is possible to make fantasy math with actual numbers just as it is possible to write about fantasy worlds with words meaning actual concepts.

Michael

I don't understand a couple of your points but I will answer what I can. One view of numbers (Korzybski's) is that they represent unique, specific, exact, symetrical and asymetrical relations. He also defines mathematics as a language capable of expressing multiordinal relations and structure in an exact form. The word 'exact' is very important in the definition since this is only possible in a language that allows no other characteristics except those found in the definitions of objects.

So a 'circle' is defined as 'the locus of points equidistant from a point called the center', for example. There is no such thing in nature, but there are things that look like circles and yes, the idea of a circle certainly came from these.

I couldn't agree more that math is not divorced from 'reality' as I indicated above, I believe it is a language closely related to how we actually perceive 'reality'. It may be possible to teach math without counting, one could start with group theory, which formalizes all the various number systems without any mention of counting.

[Michael Stuart Kelly]

Tom,

Since you are not familiar with Objectivist epistemology, here is a brief explanation as to the why of my questions. In Objectivism, all conceptual knowledge is, at base, derived from actual experience. Math is one form of knowledge and it also, at base, is derived from actual experience.

Of course there has to be a mental capacity to integrate sensory data into units, at first called "percepts" when an entity (or attribute/action/relationship) is given a specific identity, then these integrated into concepts. Concepts can further be integrated into higher concepts. This is called abstracting from abstractions (and this is what math does). So there is a faculty for doing this (mind/brain). But it needs to be fed.

Since individual knowledge is the issue, Rand based the learning of knowledge in an individual human being on developmental psychology. She started by theorizing how an infant starts organizing his awareness of his surroundings and she built it up from there.

With math, you have a brain able to develop a system from infancy on up and you have what that system is based on (i.e., experience). That is why counting actual things is considered the base of math.

Do you think it is possible to teach Korzybski or group theory to a 3 year old and skip counting toes and fingers (or other perceived things) in order to learn math? Is it possible to teach pure abstraction divorced from perceiving things to a 3 year old? The idea is not conceivable. This early base is how Objectivism accounts for the fact that math and concepts are tied to reality.

About exact systems, the ability to develop them is one of the features of an abstraction faculty (mind/brain), and not just with math. For example, the letters of the alphabet are exact and unchanging within a culture. A is A, so to speak, and will never be B in spelling words. Rules of games (like Monopoly and so forth) are unchanging also. There are many man-made constructions like this where absolute exactness is possible. This can be called projected knowledge or abstract system or any number of terms. (This is what is meant by analytic.)

In actual knowledge (concepts), definitions are contextual and new information can always be added when it is perceived or deducted. (This is what is meant by synthetic.) What does not change is what was learned before. This can be overturned and replaced, but never erased as if it never existed. One builds on this knowledge base, even if it means replacing a good part of it.

In the analytical/synthetic dichotomy, the assertion is that analytic "truths" are somehow divorced from reality because exactness is possible, whereas in other kinds of knowledge (based on observation, i.e., synthetic knowledge), exactness is impossible (or improbable). The Objectivist theory rejects this by pointing to the base connections to reality of abstract systems. The Objectivist theory does not deny the exactness of rules in abstract systems. It merely denies that the existence of such rules divorces the abstract system from reality.

In other words, adherence of an abstract system to internal rules of consistency is not "truth" as in the meaning of a concept corresponding to an external fact. The analytical/synthetic dichotomy is a form of saying that "truth" means the same thing in both cases, when it actually means two different things: (1) adherence to man-made rules within a man-made abstract system for analytic, and (2) correspondence of abstraction to reality for synthetic. Once a rule is set, there is nothing more to learn about it. You simply obey it. With facts from external reality, there is always something more to learn. Interchanging the meaning of "truth" with both in order to divorce "analytic truth" from reality is a logical error.

Michael

[Dragonfly]

If you disagree, I am all ears. What is the "essential" I am missing?

My point was that the fact that learning mathematics starts with an empirical procedure like counting fingers doesn't imply that mathematics itself is empirical. As a metaphor I used the analogous examples of a science like astronomy that grew out of astrology or chemistry that grew out of alchemy. The analogy is that a characteristic of the earlier phase doesn't have to be characteristic of the later, mature phase. I don't understand that urge to put the reality label on mathematics; we could as well say that while astronomy has its roots in mystical notions, astronomy has mysticism in the background, suggesting that mysticism is somehow part of astronomy. Your attitude is similar to that of those who think that the meaning of a word should be determined by its etymology.

Then you ask: "You equate learning to count from fingers and toes with astrology?" Sigh. You don't seem to understand the function of an analogy. At the moment I'm translating a French book about sand, in which the mechanism of the sandglass is explained. For a description of the behavior of the sand grains in the narrow part of the sandglass, the writer uses the metaphor of two roads merging into one, comparing the behavior of the sand grains in the narrowing to that of cars coming from the two roads at the junction, resulting in a congestion, where single cars only intermittently enter the single road. No doubt you'd ask now: "You equate a sand grain with a car?!", which would miss the point completely, as the essence of the metaphor is not the similarity of a sand grain with a car (there isn't much similarity, this is the inessential part of the metaphor), but of the similarity of the behavior of sand grains in a constriction and the behavior of cars in the merging roads (which is the essential part of the metaphor). In the same way the essence of my example of astrology was not the similarity of counting fingers with astrology, but the similarity of a system outgrowing its origins.

My god, why am I wasting my time on all this...

[Michael Stuart Kelly]

Dragonfly,

Astrology is not something all children go through to learn science. It was a development from primitive cultures (meaning adults). Counting is something all children have to do to learn math. That is why it cannot be discarded while astrology can be. Astrology is proven to be false as scientific knowledge. Counting is not proven to be false as math.

One is a very poor analogy for the other. These are pretty essential considerations.

A person discards the false because it is not knowledge. You wish to discard the true because it is not convenient to your theory.

Michael

[Dragonfly]

You still don't get it. I give up.

[Michael Stuart Kelly]

Dragonfly,

Sorry. I just can't divorce math from reality. There is a reason why it works so well with reality. It is based on reality.

And I fully understand that following unchanging rules is part of a man-made system of abstraction.

Michael

[BaalChatzaf]

Dragonfly,

Sorry. I just can't divorce math from reality. There is a reason why it works so well with reality. It is based on reality.

And I fully understand that following unchanging rules is part of a man-made system of abstraction.

Michael

Some mathematical structures are well matched to physical reality. Some are very abstract and not matched at all.

Ba'al Chatzaf

[Brant Gaede]

Michael, Objectivist epistemology can't be a tool invalidating certain types of mathematical abstracting unless it is in itself part of a mathematical system and can be used also to validate the abstracting (if it is validatable). It's not math. I would think the mathematician would ask what the abstracting accomplishes and why is it valuable and correct. I'm way over my head here, but might not an example of mathematical abstracting today, valid only unto itself, be used later in a way making it valid unto something else (even if only to another mathematical abstraction) and finally, eventually, valid unto your sacred "reality"? After all, what does Objectivist epistemology particularly refer to--nothing actually--other than itself? Reality is poured into the epistemology. The epistemology is not poured into reality. It does not become part of reality; neither does any math including arithmetic even though concrete reality is right next door to arithmetic.

--Brant

[Michael Stuart Kelly]

Some mathematical structures are well matched to physical reality. Some are very abstract and not matched at all.

Bob,

Where have I said differently?

Still, at the base, there are numbers. If you have numbers, you have mental units that were developed by counting things.

There are conceptual structures that are not matched to reality at all. Look at the unicorn example that is always provided. Just because we can mentally create make-believe, does that mean that concepts per se are divorced from reality because they can be employed in make-believe? Of course not.

But that seems to be the argument for math I keep reading. Why can't there be make-believe in mathematics? What do you call a logical projection divorced from reality?

I call it make-believe.

Michael