An Viable Alternative To Platonism In Math?

[LauricAcid]

Definitions are my Achilles heel, but off-hand the best I can give is: evaluation by comparison to an objective standard.

If that's all you mean by measurement, then I don't see nearly as much difficulty with your view. But, though I don't have the text in front of me, the OPAR definition includes that the comparison is of concretes and that it is quantitative [my emphases]. On the other hand, if we tightened your definition just a bit more to 'evaluation by objective standard' then I VERY much agree that this is at the heart of mathematics (but while at the heart, still not providing a full definition of 'mathematics' objectivity of evaluation is a necessary though not sufficient condition for mathematics). And I don't take that as a definition of 'measurement' - neither as a common defintion nor an Objectivist definition. Edited December 13, 2005 by LauricAcid

[LauricAcid]

Mesaurement-omission is key, but ask yourself what the ultimate goal is, in omitting measurements in mathematics. The assumption is that, at some point, you will put the omitted measurements back in again, for the purpose of arriving at some new measurement you were unable to make without mathematics.

In some instances yes and in other instances no. And not only is measurement often omitted, it is not even a consideration to begin with. Though, this remark is apropos measurement as I thought it was being used in an Objectivist sense, which includes that measurements are quantitative and of concretes.

I think you misunderstand me. I don't regard set theory as a foundation of mathematics;

No, I wasn't claiming that you assent to set theory as a foundation. My point was that no matter how you personally feel about set theory as a foundation, one cannot even function in mathematics as it exists without set theory. By this I don't mean that other foundations are not possible, but rather that, as it now stands, it would be virtually impossible, utterly impractical, to study mathematical logic, algebra, analysis, topology, probability, graph theory, et al., without dealing with sets at nearly every turn of the page.

It's a mistake to separate the function of any subject from the subject itself.

You're begging the question by holding that the function of mathematics is its technological application. That is one of the very important functions, and surely at the core of why humans took upon mathematics. But it is not the entirety of the motivation, function, purpose, or benefits of mathematics, especially since so much mathematical endeavor has had as its motivation and aim intellectual satisfications and adventures onto themselves.

Who says all measurements are quantitative?

Hold that thought. Let me get an exact quote from Objectivist text. Edited December 13, 2005 by LauricAcid

[LauricAcid]

""Measurement," writes Miss Rand, "is the identification of a relationship - a quantitative relationship established by means of a standard that serves as a unit.

The process of measurement involves two concretes: the existent being measured and the existent that is the standard of measurement. Entities and their actions are measured by means of their attributes, such as length, weight, velocity. In every case, the primary standard is some easily perceivable concrete that functions as a unit. One measures length in units, say, of feet; weight in pounds; velocity in feet per second." - OPAR, pg. 81 [bold added]

"[...]In both cases [measurement and conceptualization], man relates concretes by the same method - by quantitative means." - OPAR, pg. 82 [italics original]

Edited December 13, 2005 by LauricAcid

[dondigitalia]

If that's all you mean by measurement, then I don't see nearly as much difficulty with your view.

That is what I mean when I use the term "measurement."

No, I wasn't claiming that you assent to set theory as a foundation. My point was that no matter how you personally feel about set theory as a foundation, one cannot even function in mathematics as it exists without set theory. By this I don't mean that other foundations are not possible, but rather that, as it now stands, it would be virtually impossible, utterly impractical, to study mathematical logic, algebra, analysis, topology, probability, graph theory, et al., without dealing with sets at nearly every turn of the page.

I don't disagree with you here, for the most part. Set theory is the foundation of a lot of higher mathematics, but I've heard it asserted (maybe even earlier in this thread; I'm not going to go back and look) that it is the necessary base of mathematics as such. I've heard it asserted that, since we can describe numbers and he relationships between them in terms of sets, that we actually derive numbers from sets, completely ignoring the fact that we knew of numbers long, long before set theory was around.

But it is not the entirety of the motivation, function, purpose, or benefits of mathematics, especially since so much mathematical endeavor has had as its motivation and aim intellectual satisfications and adventures onto themselves.

Here, are you talking about some pursuing knowledge for its own sake? I don't deny that some people do this. If that's what you mean, I'll just say that I emphatically disagree with that view of pursuit of any sort of knowledge, but that's a discussion of ethics, and is completely off-topic.

""Measurement," writes Miss Rand, "is the identification of a relationship - a quantitative relationship established by means of a standard that serves as a unit..."[...]In both cases [measurement and conceptualization], man relates concretes by the same method - by quantitative means." - OPAR, pg. 82 [italics original]

Those quotes are ineresting, and I'm going to have to read them in full context. (I do remember the passage from ITOE now hat it's been quoted back to me. Of course, Objectivism (rightfully) sets the context for terminology in any discussion here, and I can accept that I've been using a word to denote a different concept than Ayn Rand and Peikoff. Definitions are a huge weak point of mine, so I'm not entirely certain I even gave he most accurate definition I could have (although it is one of my better ones). I'll hold to it for the time being, and accept that I'm probably at fault for muddling things. I'm always willing to change my terminology when there's a confusion like this. I'll put the ball in your court on that one, although I'm not sure there's much left to discuss if my definition of "measurement" alleviates your concerns about my position.

[LauricAcid]

I don't disagree with you here, for the most part. Set theory is the foundation of a lot of higher mathematics, but I've heard it asserted (maybe even earlier in this thread; I'm not going to go back and look) that it is the necessary base of mathematics as such. I've heard it asserted that, since we can describe numbers and he relationships between them in terms of sets, that we actually derive numbers from sets, completely ignoring the fact that we knew of numbers long, long before set theory was around.

I recognize that there are other foundational systems, so I don't claim that set theory is the only one or even necessarily the best one. I'm just saying that, as the chips have fallen, one can't get through even basic texts on mathematics without sets and some context of set theory. As to numbers having been discovered before set theory, I don't see that as a good objection to putting sets before numbers in an axiomatization. Discovery and initial conceptualization is historical and anecdotal. Axiomatization is post facto conceptualization. Two different things.

I'm always willing to change my terminology when there's a confusion like this. I'll put the ball in your court on that one, although I'm not sure there's much left to discuss if my definition of "measurement" alleviates your concerns about my position.

It doesn't completely eliminate my questions or objections, but it makes them a lot less pointed, that's for sure. As to terminology, I have no axe to grind here about the word 'measurement'. If you like to stipulate that you're using it as defined by Objectivism or that you're using it in some other way, then I have no quarrel with your usage as long as I know what it is. Thanks for your remarks.