aleph_0

The only evidence I know of that implies the existence of genuine mathematical objects is the fact that every attempt to explain it appears Platonic, even when attempting to avoid it. I am here looking to find a sensible non-Platonic answer. Arbitrary: 1. based on personal whim: based solely on personal wishes, feelings, or perceptions, rather than on objective facts, reasons, or principles 2. randomly chosen: chosen or determined at random 3. law not according to rule: based on the decision of a particular judge or court rather than accordance with any rule or law 4. authoritarian: with unlimited power 5. mathematics assigned no specific value: used to describe a constant that is not assigned a specific value Microsoft® Encarta® Reference Library 2005. © 1993-2004 Microsoft Corporation. All rights reserved. The Axiom of Choice does not produce arbitrary results--the reals that it assigns are the same ones each and every time, it is not dictated by any whim, there is a law that prescribes them you just can't define it. Do you know what the term 'arbitrary' means when applied to a subject?

Not knowing the higher levels of mathematics hasn't prevented me from understanding Frege's The Foundations of Arithmetic, Benacerraf, and Carnap (and your statement about method is, granting that you make no distinction between "method" and "system", liable to the same criticism that Benacerraf is liable to). I see what you mean by "infinity isn't an object, it's a method" I just don't see how it will, if indeed it can, avoid Platonism. And I have read a rigorous definition of limits, which I understand, though I haven't yet worked with them.

Hal, are you sure we haven't found a mathematical object? What evidence is there that the Zermelo sets don't describe such an object? And were we not to have found it, that does not mean it cannot exist. Maybe it cannot exist for some other reason, but not that one. And yes, you don't need the AoC to prove magnitudes of infinity, and in fact I did not use it to do that (as you'll note, the proof of magnitudes of infinity assumes the undecidability of the decimals in a real number, which distinguishes it from a rational). But the proof of magnitudes demonstrates the nature of the AoC--that there is some function that is not definable but which orders the real numbers, *is* the property that makes them larger. "If you reject infinite sets altogether, then AC isn't an issue, you can prove theorems to take its place." True enough, punk, but infinity is rather useful and it's hard to understand the quantity of natural numbers any other way. "If it doesn't make sense you have to pause and think hard: Is my issue really just tantamount to me being concerned that I can't prove the axiom as a theorem?" No, I have no problem about proving it. I have a problem about what it implies. "but you don't hear much about the Axiom of Infinity (AI) (which, as I indicated above, is the thing that causes us to think we might need AC)." I brought it up. "Saying that the concept of '4' is even, would be like saying that my concept of 'unicorn' has a horn. But this is obviously absurd - the concept is inside my skull, probably represented as a neuronal pattern in my brain." Well this runs into the inherent trouble/contradiction of reductionism that Kripke pointed out in On Sense and Reference.

We should. What score did you make? I haven't taken mine yet, but I'm currently testing around 160. With a bit more studying, I'm hoping I can average in the 165-170 range. It's all a matter of about one or two questions per section for me, which is frustrating because I lose all my points from the time-limit. You need at least a 164 (or about the 90th percentile) to teach LSAT through Kaplan. I made that on my GRE and SAT, so I'm definitely tutoring those. Since I was a philosophy major, I figure I could probably do the LSAT too.

I haven't taken the normal progression of math classes because every time I try to sit through a normal lower-level math class I have to fight to control projectile vomiting. I hate having to memorize formulas like catechisms, so I never took much math in college. I've taught myself up to trig by actually learning concepts, and I'm just now finishing that and broaching calc. But I have approached the rest of math through the backdoor via symbolic logic, mathematical logic, and philosophy of math. Eventually I hope to teach myself through to set theory, proofs, complex numbers, and a little application in stats, probability, and some (or a lot, depending on how I like it) physics.

Well I tried that maneuver too, but the problem is you're left with a "method". So what is the ontological status of this method? Another thing that seems like a Platonic form? And why is it that mathematics today treats infinity like a true object, and gets quite useful results in doing so? And I'm sorry, but I don't understand your answer to the Axiom of Choice. You don't put anything into it. It puts very defined, un-arbitrary results into the real number-line. You simply cannot produce an algorithm that will produce all real numbers. Which implies that some other function does it--an abstract function that sounds like a Platonic form. Thank you and good to meet you.

Hey, I used to go to FSU (for a year). And I will soon be tutoring the LSAT. We should talk.

Hi. I'm Adam, from central Florida, a graduate of Brandeis Uni, now a tutor for Kaplan until I get into grad school. I study philosophy , history , and increasingly more mathematics ... which means I am the model from which all nerds are forged. All the best, Adam

I am curious if there is any student of objectivism who knows about abstract math and metamathematics, and perhaps philosophy of math. For more than a year now I've been wrestling with the problem of what is a number, particularly, how does one account for the infinity of numbers. In a related question, how does one account for the very spooky Axiom of Choice? Both of these concepts seem to imply a kind of Platonism. If anybody is familiar with this topic, I'd appreciate it. If anybody wants to look into it, the first link below is something I've already written on another (slightly dead) objectivist forum at MySpace that covers the question of infinity. The second link is my rough-and-ready discussion of the Axiom of Choice. But if you're not familiar with the philosophy of mathematics, it may be hard to provide insight. http://forum.myspace.com/index.cfm?fuseact...299279DD3791507 http://forum.myspace.com/index.cfm?fuseact...299279DD3791507