Posted on 10/25/2004 1:46:25 AM PDT by accipter
CONSIDER a verbal description of the effect of gravity: drop a ball, and it will fall.
That is a true enough fact, but fuzzy in the way that frustrates scientists. How fast does the ball fall? Does it fall at constant rate, or accelerate? Would a heavier ball fall faster? More words, more sentences could provide details, swelling into an unwieldy yet still incomplete paragraph.
The wonder of mathematics is that it captures precisely in a few symbols what can only be described clumsily with many words. Those symbols, strung together in meaningful order, make equations - which in turn constitute the world's most concise and reliable body of knowledge. And so it is that physics offers a very simple equation for calculating the speed of a falling ball.
Readers of Physics World magazine recently were asked an interesting question: Which equations are the greatest?
Dr. Robert P. Crease, a professor of philosophy at the State University of New York at Stony Brook and a historian at Brookhaven National Laboratory, posed the question in his Critical Point column and received 120 responses, nominating 50 different equations. Some were nominated for the sheer beauty of their simplicity, some for the breadth of knowledge they capture, others for historical importance. In general, Dr. Crease said, a great equation "reshapes perception of the universe."
(Excerpt) Read more at nytimes.com ...
No, they each paid 10$ and the clerk kept two, refunded each one and the 25 dollar balance(30-5) matched the bill.
3*8 = 24 ... $1 left to tip the clerk
As long as both sides are actually equal, then the statement is true. Both sides must represent the same thing, or object.
" The question is whether "A is A" has any general axiomatic meaning, and in the example of the six-sided sphere we see that it has no meaning. IOW, whether "A is A" has meaning outside of an abstract concept, depends entirely on what we choose to plug into "A."
The beauty of this equation is that it is the simplest statement of truth. An object is what it is and nothing else. It represents fundamental truth about the object, regardless of whether, or not the truth is a falsehood. Take for instance your 6-sided sphere...
6ss = { r=const. for all theta and phi AND has 6 sides}
The object 6ss is defined by this equation. Both sides represent the same object. THe object however is not real, is impossible to construct and has internal contradictions, based on the meanings of sphere and side. So it is a useless object that is by it's very nature, unreal.
On the other hand, most folks are interested in real objects, most often they are useful. Normally the name, or word that represents the object occurs on the lefthand side. The definition, or set of quantifications for the object occur on the right side. As long as those quantifications are true about the object, then the statement is true and that is what is important.
Four = 4, or epi*i =0 are examples of a = a. Here's another one:
marriage = {A contract union of one man and one woman, contracted by mutual agreement to exist permanently as life allows.}
As you can see, marriage is not a pen, a can of pop, or 2 men. The representations for the object, "marriage", are both true and real. No other quantification fits. Had you noticed, that's all Rand said also. She commented that others worked with a != a. ie the 'toon with is != is, or sex != a monica, or SCOTUS in an extended scheme asserting that the right to privacy trumps legislation and the right to life. 4th Amend = {Protection of the right to privacy against fishing expeditions. It requires witness, reasonable cause, and a warrant. The Amend. does not provide for protection of anything done in private from legislation, or sanction.}
4th Amend(Warren court) = {Establishment of an inalienable right to privacy, that trumps the right to life and prohibits legislation regarding anything done in private, but only in the case of abortion at this point.}
The inherent and self evident truth in the 4th is given in the first eq(example of a = a). The second equation is an example of a != a). SCOTUS presented the second as a true equation. It was however, false, both sides of their eq. contained quantifications for different objects, not the same object.. THat's why it is only true with "!=".
You have: (a) conceded my point; and (b) raised a totally different and irrelevant issue.
I had no intention of arguing the details of what some individual "A" might be. Such details are pointless in this context. I only wished to say that whatever "A" may be in any particular scenario, it is what it is. That's what "A is A" means. And it's axiomatic.
... it's not enough to just say "A is A" and think you've done something swell.
I don't know about the "swellness" of it, but it's a basic statement about reality. No big deal for those who grasp it, but a ghastly flaw for those who don't.
It's quite clear that not all "A's" are created equal ...
Yeah, so? The axiom says only that each individual "A" is equal to itself. Why is that an issue for you?
Two things not equal to themselves are not equal to another thing either, necessarily.
... are equal to each other, however.
A=A is really a definition of what is is, is it not?
i see too uh nuance here. the orogonal equation is simple enough to aply to most real life situation, and these invole 99.9% of applictions used by engineers. when broadened to the einstein universe i understand the dfferences.
Pretty much, yes. One could easily go on for pages, expanding on the nature of existence, reality, identity, etc., but in a nutshell ... yeah, you've got it.
Interesting to note that all well-defined forms of "is" (equivalence relations) are, by definition, reflexive (in addition to being symmetric and transitive.)
Thus, "A=A" follows by definition, regardless of what "A" might be.
Or so it seems to me.....
"Walter Scott is the author of Waverly"
Epimenides is a Cretan.
Fair enough.
But in the case at hand, "A=A," it pretty clear that Rand wasn't saying that "A" was an element, or subset, of itself, though Russell might.
;-)
a) No. I defined my terms carefully. I used "axiom" in the sense of "suppose, for the sake of argument," which is the second, and weakest, definition of the term in Webster's dictionary.
b) I was raising an issue that you said was not important -- but it is important, if "A is A" is to have any meaning.
Yeah, so? The axiom says only that each individual "A" is equal to itself. Why is that an issue for you?
It's not an "issue" for me -- it's just a fun discussion where I'm trying to take a contrary view, and seeing where it goes.
However, the point remains that you can plug any old thing into "A is A" and be logically correct, even though "A" itself may make no sense whatsoever. You claim that it's a "basic statement about reality," but it's not -- it's only an abstract concept until you actually plug "things" into "A."
For grins, let's plug "PatrickHenry" into the equation. Thus, "PatrickHenry is PatrickHenry." Fair enough, but does that tell us anything? No, not really, because we don't know what "PatrickHenry" really is. Physically you're not the same NOW as you are NOW. IOW, (PatrickHenry(t) != PatrickHenry(t+dt)). The question is: which of those is "PatrickHenry?" Are they both "you" (in which case you've violated "A is A")? Or is "PatrickHenry" a continuum over time, in which case "A is A" means something entirely different from the way you've stated it ("each individual A....")?
If you look carefully, it would seem that there is a fundamental difficulty of our having to define "A" before we can actually use "A is A." We seem to be a necessary part of the equation. More on this below.
I only wished to say that whatever "A" may be in any particular scenario, it is what it is. That's what "A is A" means. And it's axiomatic.
Let's look at that a bit more closely, and see if it's really self-evident (which is the dictionary sense of "axiomatic" you're using).
What you're really saying is, that the existence and characteristics of a thing are independent of any observer. Yet this is apparently not certain -- see, e.g., the Copenhagen Interpretataion of quantum physics, or the Theory of Relativity. From the link:
In general, Bohr considered the demands of complementarity in quantum mechanics to be logically on a par with the requirements of relativity in the theory of relativity. He believed that both theories were a result of novel aspects of the observation problem, namely the fact that observation in physics is context-dependent. This again is due to the existence of a maximum velocity of propagation of all actions in the domain of relativity and a minimum of any action in the domain of quantum mechanics. And it is because of these universal limits that it is impossible in the theory of relativity to make an unambiguous separation between time and space without reference to the observer (the context) and impossible in quantum mechanics to make a sharp distinction between the behavior of the object and its interaction with the means of observation.
My rough translation: What "A" is, is in part a function of the observer -- or, "A isn't really A unless somebody sees it being A." Or, as per the example above, our definition of A, is part of what A is.
The Copenhagen interpretation has always bugged me. Quantum physics seems to presuppose, for example, that the mathematics used to describe the phenomena are somehow immune to the "observer effect." If we accept that, then it appears we have to make some distinctions between things physical (which are not "A is A" entities), and things metaphysical (which can be "A is A" entities).
NEA math:
$25+$2+$1+$1+$1=$29?
b4i4q(ru/18qt{pi}).
you engineers will understand the significance of this; and who says that science is amoral?
Now my favorite equation is along the lines of:
f(x)=x^n,
because it is most optimitic and reflects, imho, the heart of God.
What I "see" is that you're the one defining marriage -- you're in the loop. "Marriage is Marriage" (a particular instance of "A is A") has meaning only in the context of your definition of marriage. The same goes for "unalienable rights" of various descriptions.
It boils down to whether or not things have meaning outside our definitions of them, or the context in which we place them. In areas such as marriage or rights, it's question of whether our moral computations are objective or relative.
The problem you run into in trying to resolve that question is that you can't really achieve "objectivity" without assuming action by an agent "outside" your point of reference -- God, so to speak. If one accepts the existence and action of God as axiomatic (e.g., the great "I AM" of Exodus 3:14), then objectivity is presumably possible within any human context. When one relies on personal definitions, then objectivity is impossible.
The Founders understood this logic, which is why they stated that we are "endowed by [our] Creator with certain unalienable Rights...." The Warren Court, and modern culture in general, define rights in terms of current context -- what "seems right at the time," as defined by political processes.
A cardioid?
i think it's a turbocharged parabola, isn't it; the octane being n?
Nope i simply observe and state what I find.
" The problem you run into in trying to resolve that question is that you can't really achieve "objectivity" without assuming action by an agent "outside" your point of reference -- God, so to speak."
Nope. Any rational and honest being can observe what exists and identitfy it. All my not be know at any particular point, but there are no inherent limitations to what can be observrved and known. "When one relies on personal definitions, then objectivity is impossible."
What is, is.
" If one accepts the existence and action of God as axiomatic (e.g., the great "I AM" of Exodus 3:14), then objectivity is presumably possible within any human context.
Objectivity does not depend on God. It depends on the person doing the observation only.
It is also impossible to accept and know the God of Exodus as axiomatic. He can not be observed . All you have is the word of other men. In addition you were told by God Himself that no one could know the Father, except though Jesus, who is the Father. The only way to know God is by what Jesus said and the sign of the resurrection, which is the directly and personally observable Holy Spirit.
See "Quantum Mechanics", sub-reference "Heisenberg."
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.