iQuestor wrote:Malik - I said your comments were arrogant, not you.
Oh hell, you're right on both counts. Both myself and my comments are arrogant. It's something I need to work on, rather than pretending to be proud of it. And you're right about this one, too:
You may have issues, but . . .
I'd like to hear your response on the paper, but otherwise, for this topic, I am done unless someone else has a question or something to add.
Well, I read parts of it in detail, and then skimmed other parts. I was pleased to see him talk about Bertrand Russell's monumental work, and his knowledge that Russell's goal wasn't entirely achieved. This gave me hope that he might actually address Husserl's points, too. But that wasn't there. This issue is indeed closely related to the problem of universals, and many have described Husserl's position as Platonic (i.e., that universals have some real, objective existence beyond the physical plane). I've always thought Plato's view of universals was a bit too metaphysical and unjustified, until I came upon Husserl's powerful argument against psychologism--another part of the issue not touched by this paper.
It's difficult to evaluate Lady's position--or rather, why you think it supports your own view. He claims that numbers and math have an objective existence (though not tangible). This is what I mean by calling them ideal objects (rather than physical objects). They have an existence which transcends our invention of them. Though they don't exist as physical objects, they are still objects we discover, rather than invent.
The relevant portions are here:
By means of a method known as the Cantor Diagonalization Principle it can be shown that any set of computer programs specifying real numbers will leave infinitely many out, even if the set contains infinitely many programs, One can apply this to the set of all computer programs which specify real numbers and see that there must be infinitely many real numbers which are not specifiable by any program. It doesn't make any difference what computer language the programs are written in. In fact, the same is true if instead of computer programs we choose to specify the decimal expansion of a real number by means of a paragraph (or even a book) written in English.
We now show how to produce a decimal expansion which is not in our set . . . .In terms of the issue we are discussing, this is quite disturbing. We have argued for the existence of natural numbers, and even rational numbers, by arguing that the natural numbers are deeply rooted in our language and thus in our thinking and that even though numbers have no tangible existence, they do describe very real situations in the world. But real numbers, being infinitely precise, do not accurately describe the world. Furthermore, not only does human language not contain words for most real numbers, but it is not possible to label them even using such contrived tools as computer programs. (No system for labeling the real numbers using finite strings of symbols take from a finite alphabet, no matter how huge, can possibly work.)
So, if math and numbers are merely human concepts that we invent, how can there possibly exist numbers which no possible labeling or symbolic system can symbolize? Doesn't that mean that the numbers exist in a manner that goes beyond our labeling and symbolizing? That they exist even before we've named them?
He says this is disturbing because he was trying to build a case that the "existence" of numbers and math are completely dependent upon our language--merely an invented tool for describing the world, as you have argued. But here he admits that there are aspects of math which not only transcend our language, but also render them inappropriate for describing the world (these properties being their infinite precision). But if we truly
invented them for describing the world, then how can they have these unforeseen properties which aren't mirrored in the tangible world? Why would we put these inappropriate properties within our tools?
Obviously, we didn't invent those properties. Numbers already had them, and we discovered them.
However, I'm puzzled because by the end of this paper, Lady seems to come to the oppposite conclusion supported by the evidence he has amassed: that numbers and infinity are invented. He admits that the source material (Lakoff and Núñez) took him a long time to understand due to his assumption that numbers are inventions, and then asks us to trust his paraphrase of their conclusions. But if his presuppositions are what caused his diffuculty in understanding Lakoff and Núñez, then how does it make sense that they were saying the same thing he always assumed? Shouldn't his assumptions make them
easier to understand? I'm not convinced that he ever settled his confusion over their position.
I think this is best summed up early on when he says:
Certainly my own knowledge of philosophy is at best that of a dilettante.
While I can't claim that I'm any better than this guy, I certainly wouldn't trust his cursory evaluation of the philosophy of mathematics over Husserl's.
Let's look at a few more of your statements.
Iquestor wrote:We can never express Pi as a completely accurate number, no matter how hard we try.
Does this mean that the ratio of the circumference to the diameter of a circle doesn't exist? If numbers and math only exist as invented expressions, what about mathematical truths which we can't express, even with an infinite amout of invented symbols? Clearly, there is a reality--a truth--contained here which our invented symbolic description cannot capture. But that truth is no less mathematical just because we can't fully describe it.
This is the point Wayfriend was making about Godel. [Excellent book, by the way.] Godel proved that there exist true statements within math or logic which--though we know they are true--cannot be proved within the invented system we use to symbolize math.
Wikipedia says:
These theorems show that there is no complete, consistent formal system that correctly describes the natural numbers, and that no sufficiently strong system describing the natural numbers can prove its own consistency. These theorems are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible.
And:
Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false. Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove.
So mathematics exists "beyond" our ability to capture or describe with axiomatic or formal systems. Godel proved it. We can comprehend it, but we can't symbolize it. Think about that. The mind can comprehend mathematical truths which it cannot symbolize. Clearly, math is more than our symbols. Its existence transcends our symbolic powers (or even that of an infinitely powerful computer running
forever). Math is not in the symbols. Our symbolic language--which we did invent--is used to describe a pre-existing mathematical "world"
just as it is used to describe the tangible world. In other words,
the ideal or mathematical/logical "world" exist just as independent of our symbols as the tangible world exists independent of our symbols. (But that doesn't mean that numbers are tangible).
This brings me to an important question regarding your view: what is the difference between knowing and describing? You keep saying that we can ONLY describe the world with math, but that this mathematical order doesn't exist in the world. Does this mean that we can't know the world, we can merely describe our experiences of it? Do we reach any truth with mathematical descriptions? Does your view collapse into solipsism? If so, we're back to faces in the cloud--a point I think you've consistently misunderstood. Let's look at your response to this issue.
I wrote:
If quantity and mathematical structure don’t really exist in nature, then how is this description different from a face in the cloud? You've said that mathematical relationships, too, are products of the human mind. So what's the difference? If it’s just order that we impose upon the system, that’s exactly the same as what you’ve described above [in refuting the existence of cloud faces].
You responded:
sorry. A face doesnt exist in a cloud. it exists in the observers mind. you will never convince me othewise.
You seem to think I'm arguing that faces in clouds actually exist. Maybe you think I'm aruging this because I'm arguing that numbers are real (which you believe are just as imaginary as cloud faces). Let me be clear: I'm not arguing that cloud faces are real. I'm arguing just the opposite: faces don't exist in clouds. HOWEVER, based on your characterization of mathematics, one must conclude that the order we intuit everywhere in the universe, like the structure of gravity, doesn't really exist in the universe, but instead in our heads. There is no difference in what you're describing, and my example of a face in the clouds. You cannot argue that mathematical descriptions are inventions without rendering that which they describe as an invention, too. Is the earth's orbit an ellipse, or not? Does it merely
look like an ellipse (in the same way water vapor
looks likea face from a certain angle), or does it retain this shape even in the absence of humans to view it? Does it suddenly move in a straight line, or something other than an ellipse when we're not here? [Now granted, it's not a perfect ellipse due to gravitational perturbations of the moon and other bodies--but these perturbations can all be accounted for with mathematics, too. In other words, the discrepancy between a perfect, mathematical ellipse and the actual shape isn't due to a limit of math as a descriptive tool, but due to physical complications
which can be accounted for with that math, and thus included in that description.]
In summary, you seem to think "description" precludes "knowledge." I'm arguing that even though description is taking place, objective knowledge is still being achieved--even though that knowledge is paradoxical. Indeed,
achieving objective knowledge through our subjective perceptions is itself the biggest paradox of all. But the alternative is solipsism. If you don't accept the paradox, then that's where your views end up. You are the only thing that exists.
