An Viable Alternative To Platonism In Math?

Science & Technology Physics and Mathematics
41

I think that you are including both generalization and instantiation in your concept of “induction”. Most of us consider instantiation to be deduction, not induction.

The act of applying the rule/law/theorem to an instantiation is inductive. In this context, I am using “applying” to mean: deciding that a specific problem is best solved using a particular rule/law/theorem, and initiating it’s use. The instantiation which follows that induction almost always involves deduction, but oftentimes involves further inductions as well.

42

First, I think I know what book you are reading. It is Brown’s “The Philosophy of Mathematics: An Introduction to Picture and Proofs”. This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

First, let’s understand that Rand was not a scientist and that her ideas need to be understood in the context that gave birth to them. She once said that the amount of matter in the universe is fixed: This statement is patently false. Mass-energy is constant, not mass by itself. She clearly thought in Newtonian terms that have been shown inaccurate (I will not use the word false because they are TRUE IN MOST CONTEXTS. Rand herself writes about the context dependence of truth and the need to reformalize thought in the light of new information.

Mathematics is not the “science of measurement”. This would completely eliminate number theory, the theory of polynomials, and much of analytic geometry from mathematics. Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies. It is foremost the study of identity—the internal consistency of the structures we use to discuss the outside world just as Grammar is the study of the way language is used to discuss the outside world. This analogy also holds who Math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in “The Principles of Mathematics Revisted”). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

First, I think I know what book you are reading. It is Brown’s “The Philosophy of Mathematics: An Introduction to Picture and Proofs”. This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

First, let’s understand that Rand was not a scientist and that her ideas need to be understood in the context that gave birth to them. She once said that the amount of matter in the universe is fixed: This statement is patently false. Mass-energy is constant, not mass by itself. She clearly thought in Newtonian terms that have been shown inaccurate (I will not use the word false because they are TRUE IN MOST CONTEXTS). Rand herself writes about the context dependence of truth and the need to reformalize thought in the light of new information. She had little experience with mathematics, and her views represent the view of math that was dominant in the mid 1800’s (probably the view held by her Professor Lossky). It is an incomplete view, esp. since the study of “measurement” entails many questions about the nature of space, number, set, function, and logic. To study “measurement” by itself would be to cut mathematics away from its basic context without reason or practical gain.

Mathematics is not the “science of measurement”. This would completely eliminate number theory, the theory of polynomials, and much of analytic geometry from mathematics. Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies. It is foremost the study of identity—the internal consistency of the structures we use to discuss the outside world just as Grammar is the study of the way language is used to discuss the outside world. This analogy also shows how math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense. It is inspired by experience but relys equally on the observation of the structures of representation itself.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in “The Principles of Mathematics Revisted”). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

43

Thanks for mentioning Hintikka, Franklin.

For other readers of this thread, I just mention that Hintikka, one of the giants of mathematical logic, issues a bold and fascinating challenge. Here are a couple of links:

Notice here not only the article by Hintikka and Sandu, but also Wang’s article on Skolem and Godel, other articles about Skolem, and the article about first order logic:

http://www.hf.uio.no/filosofi/njpl/vol1no2/contents.html

Here’s a review of Hintikka’s book:

http://projecteuclid.org/Dienst/UI/1.0/Sum…rml/1081173780

44

First, I think I know what book you are reading. It is Brown’s “The Philosophy of Mathematics: An Introduction to Picture and Proofs”. This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

Was this directed to me? You didn’t quote anyone, and my post immediately preceded yours, so I can only guess. I’ve never heard of this book, so I can assure you, I’m not reading it.

Mathematics is not the “science of measurement”…Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies.

All of those things are either measurements themselves, or categories of things which must be measured. (What do you mean by “structure” in this context? The physical structure of existents, or the conceptual structure of a logical/mathematical system?) Numbers & quantity are both concepts of measurements. Space is a relationship between objects in temporal coexistence; change is the relationship between the characteristics of a single existent at different points in time. Both are things we measure.

This analogy also holds who Math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense.

Well, yeah, the same can be said of just about any science. No knowledge is entirely empirical or entirely conceptual; there is always an element of each.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in “The Principles of Mathematics Revisted”). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

I’ll thank you for mentioning Hintikka as well. I’ve had his work recommended to me a couple of times, but have never anything. I’m glad you reminded me.

45

What measurement is involved in the theorem that there exists an empty set? What measurement is involved in, for example, the union axiom? In the theorem that Zorn’s lemma is equivalent to the axiom of choice? In the theorem that every field of sets is a Boolean algebra? That every equivalence relation induces a partition? That there is always a finest partition? What measurement is involved in the definition of a lattice?

46

To say that mathematics is the science of measurement does not mean that every aspect of it can be reduced to a measurement. It is, in effect, an answer to the question: Why do we need mathematics? The answer: To measure things. It is the need to measure that gives rise to mathematics at all.

In regard to your specific questions, measurement isn’t involved directly. But what’s important to recognize is that, while extremely useful, set theory is not fundamental to mathematics as such. Ultimately, though, when applied to reality, mathematics is always measuring something.

47

To say that mathematics is the science of measurement does not mean that every aspect of it can be reduced to a measurement. It is, in effect, an answer to the question: Why do we need mathematics? The answer: To measure things. It is the need to measure that gives rise to mathematics at all.

I think that’s reasonable. But (1) I don’t see that qualification given explicitly in, for example, OPAR, (2) Even if measurement is the primary motivation, I don’t see how one can say that it is the only motivation, since so much of mathematics has to do with relations that actually omit measurement. Omission of measurement, for the purpose of greater generalization, is common in mathematics. The relations studied in mathematics are often not those that are about quantitative comparision.

But what’s important to recognize is that, while extremely useful, set theory is not fundamental to mathematics as such.

There are other foundations proposed, but among your basic mathematics textbooks, you’d have to search high and low to find one that doesn’t make use of sets, especially in such basic studies as analysis, topology, algebra, graph theory, and probability. One can choose all kinds of examples of mathematics that are about relations that are not quantitative from many areas of mathematics. I just chose those examples since they are simple ones.

Ultimately, though, when applied to reality, mathematics is always measuring something.

(1) That says something about application of mathematics more than it says about mathematics itself, (2) For example, studying Boolean algebras for the purpose of setting up, say, a switching design, is not quantitative. Example after example can be given.

48

I think that’s reasonable. But (1) I don’t see that qualification given explicitly in, for example, OPAR, (2) Even if measurement is the primary motivation, I don’t see how one can say that it is the only motivation, since so much of mathematics has to do with relations that actually omit measurement. Omission of measurement, for the purpose of greater generalization, is common in mathematics. The relations studied in mathematics are often not those that are about quantitative comparision.

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.

There are other foundations proposed, but among your basic mathematics textbooks, you’d have to search high and low to find one that doesn’t make use of sets, especially in such basic studies as analysis, topology, algebra, graph theory, and probability. One can choose all kinds of examples of mathematics that are about relations that are not quantitative from many areas of mathematics. I just chose those examples since they are simple ones.

I think you misunderstand me. I don’t regard set theory as a foundation of mathematics; I think the foundation goes deeper than that, and is also more simple. But, this being the main focus of my own study in found. of math, I’m not willing to say much more than that right now.

(1) That says something about application of mathematics more than it says about mathematics itself, (2) For example, studying Boolean algebras for the purpose of setting up, say, a switching design, is not quantitative. Example after example can be given.

(1) It’s a mistake to separate the function of any subject from the subject itself. (2)Who says all measurements are quantitative?

49

I think it would be helpful if you each defined the concept ‘measurement’.

50

I think it would be helpful if you each defined the concept ‘measurement’.

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

51

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.

52

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.

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"“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]

54

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.

55

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.