tjohnson

I agree with the nutshell except I would reword it like this; Analytic<=>Mathematics, Synthetic<=>Everything else. I think Einstein said something along the lines of this - insofar as mathematics applies to nature it is inexact and insofar as it is exact it does not apply to nature.

Interesting, it seems to me there are a number of species of animals where color must be considered an essential characteristic of it's existence - camouflage plays an important role in nature.

I think you'll find that we need to state our undefined terms, not agree on definitions. Even if we agree on definitions there's no guarantee we are speaking about the same thing. By agreeing on undefined terms we bring mutual experience into the picture and so agreement may follow. Undefined terms are the building blocks of all language.

Micheal, what did you do before you found O'ism, did you have no "logic"? What has O'ism enabled you to figure out that you couldn't before?

Let us look at an example. Suppose, as in my "experiment" above, we accept that we can't define 'point' any further and that we trust we both know what 'point' means in some context. In a natural language we would imagine the 'point' of a needle or tool, for example. In mathematics we also used "undefined" terms in our definitions like "a circle is the locus of points equidistant from a point called the center". The difference is that the mathematical point has no physical referent like the needle point. We can see and feel points. They have dimension, some are sharper than others, etc. Not so in mathematics. The mathematical point, even though similar to the idea of a physical point cannot be sensed, has no dimensions and has existence only in our nervous systems. This is what we mean when we say "all particulars are included in the definitions" in mathematics because the undefined terms refer to imaginary entities that are stripped of their physical properties and no other characteristics are allowed. This is why deductions work absolutely in mathematics but only relatively in natural language. Imagine trying to discover the value of pi using a string and a ruler. You might get close, you might get 3.14, but you are constrained by the "thickness" of your line and your approximate measurements with the ruler so you can only ever get an approximate answer. To sum up, natural language is an imperfect language in which we can speak about everything and mathematics is a perfect language in which we can only speak about limited things.

Micheal, mathematics is the ONLY language where the law of identity works.

???

OK, this is not so different from what I said - you can define with words no further. I don't understand this statement. If I define a linear function as "a function that can be written in the form y=mx+b", there is nothing we can observe and say "I mean that".

Don't get me wrong - I'm all for deductive and inductive reasoning, it's just that we need to realize the limitations of these methods in natural languages - when we speak about real, physical things, for lack of better terms. See this excellent site for an exhaustive list of fallacies. http://www.esgs.org/uk/logic.htm I don't personally use them but they may help one look for "verbal tricks" and avoid common pitfalls in language use.

I see no utility of formal logic - one doesn't need to take a course in logic to be logical. For all intensive purposes you may as well say 'reasonable' and 'logical' are synonymous, IMO.

It doesn't matter how many rules you establish you can't include all particulars in your definitions and so the rules may not always work. Here is an experiment; Take a word and define it then define a key word in your definition. Repeat this until you are defining in circles, for example defining 'point' with 'line' and 'line' with 'point', etc. You have reached the objective level and cannot define verbally anymore. Now we must have common experience-word relationships to communicate ie. I must trust that you understand what I mean by a certain term. This differs highly with math because there need not be any word-experience relationship in math, the definition is all we have.

Yes John, there IS a problem with deduction. In any natural language deductions can only ever work relatively since we cannot include all particulars in our definitions. In mathematics they can work absolutely so long as we follow the rules of deduction.

This is true in all natural languages, which was my point - and O'ism falls into the category of natural languages. Of course I knew exactly what you meant yet one cannot always be sure that is the case, unlike in mathematics where we can be sure EXACTLY what is meant. You cannot remove ambiguity from natural languages, if you succeed you are in the realm of mathematics.

My neighbor is a white person named Swan and he is not a bird, so much for no exceptions. Now what will you say? All white swan BIRDS are birds? Yes, I agree. If "logic" is about forms and devoid of content then it must be mathematics, you can't have it both ways. You can't make statements about ALL things because you can't perceive all things. Not so in mathematics. Let A be the set of real numbers in the interval 0<x<1. Then for ALL x in A, x<1.

Not sure I understand, doesn't a motion picture do the opposite, ie. take a bunch of static pictures of a continuously changing image? What I am talking about is what happens when we watch a motion picture. His major work, Science & Sanity, was published in 1933. Incidently, dividing up a continuously changing process into static "snapshots", like the camera, is what we do in differential calculus. Our brains cannot process change directly, we need to break it into static steps and words are just that - static abstractions from changing processes (mental images, feelings, etc.).

Good one.

To be is to be related. Numbers are used to represent unique, specific, symmetric and asymmetric relations. Zero and one are used to represent the symmetric relation of equality in addition, ie. x+(-x)=0 and in multiplication, x*(1/x)=1 . Other numbers are used to represent unique, specific asymmetric relations, ie. 1+1=2, 2+1=3, etc. In the context of this discussion it could be said that numbers are "concepts of relations" and this includes ALL numbers, including complex numbers.

Yes, and all of them useless verbalisms.

And I can tell you that it's ridiculous to discuss validity of inferences when we know the premise is ridiculous, but that's what "logicians" like to do I guess.

Good question. I like fiber bundles myself. http://en.wikipedia.org/wiki/Bundle_(mathematics)

I think 'logic' is a complete waste of time - a red herring. Are you saying all swans are white, black swans cannot exist, the two statements are equivalent, or all of the above? If you say all swans are white how could you possibly know that, have you seen all swans? How can you know black swans cannot exist? The statements are certainly related but they definitely are not 'equal' to each other.

Korzybski theorized that not only was mathematics similar in structure to nature but it was also similar in structure to our nervous system. It's a rather radical idea but one can imagine an example by thinking about how our eyes integrate static pictures into a continuously changing image similar to what is done in integral calculus. This may explain why mathematics does have such success in discovering relations because, after all, our consciousness is a joint phenomenon of what is around us and how we interact with it.

You seem to be saying that mathematical objects have their origins in our own perceptions and, if so, I don't have a problem with that, but in advanced mathematics there is little or no resemblance to anything we perceive. This is perhaps the most fascinating thing about mathematics, that is appears so disconnected from perceptions and yet can uncover relations that can be seen empirically.

Yes, it is unfortunate that the term 'imaginary' was used in this respect. Similarly for 'rational' and 'irrational' numbers, there are connotations from natural language that probably discourage people from learning more about the evolution of number systems.

Ah, that's more like it. You have to beat me because you have a need to. At least you're not using that 'for your own good' bullshit.