Circular Time

Productivity Member Writing
21

Most mathematicians do not grasp the significance of what Weierstrass did, including Weierstrass himself evidently, as his student Georg Cantor went on to develop his theory of “transfinite” numbers, or different so-called levels of infinity. 

Do you have any specific objections to Cantor’s work, in the sense of flaws that you can find in it? The idea of ‘different levels’ of infinity follows logically from the definition of sets, functions and cardinality; its not really something that you can argue with, unless you have a problem with those definitons. Personally I think that the mathematical notion of ‘cardinality’ is translated somewhat imprecisely into the English word ‘size’, and this is where a lot of the problems stem from. I agree that it doesnt make much sense to talk about “different sizes of infinity” in the standard English sense of the word, but talking about ‘greater cardinality’ in the sense of the non-existence of an injective function from one infinite set to another seems sound.

22

Where exactly in the appendix to Introduction to Objectivist Epistemology is the term “pre-reason” used? The only time I recall Ayn Rand using that term was in reference to a comment by Peikoff which used that term, and that was not in ITOE.

I haven’t read IOE for some time, but I’m sure I can remember a comment in the appendix along the lines of “I am talking about _rational_ mathematics, ie that which existed before Russell”, or something similar. Perhaps he is referring to that?

23

I haven’t read IOE for some time, but I’m sure I can remember a comment in the appendix along the lines of “I am talking about _rational_ mathematics, ie that which existed before Russell”, or something similar. Perhaps he is referring to that?

Your rememberance was close:

“When I say ‘mathematics,’ I really don’t mean the modern status of the science, but proper mathematics, rational mathematics.”

In that context I think she is referring to foundational issues.

But, anyway, when thinkonaut says

I was just using the term “pre-reason” in the spirit of the way it was used in the appendix of IOE

I take the part in quotes literally.

24

Do you have any specific objections to Cantor’s work, in the sense of flaws that you can find in it? The idea of ‘different levels’ of infinity follows logically from the definition of sets, functions and cardinality; its not really something that you can argue with, unless you have a problem with those definitons. Personally I think that the mathematical notion of ‘cardinality’ is translated somewhat imprecisely into the English word ‘size’, and this is where a lot of the problems stem from. I agree that it doesnt make much sense to talk about “different sizes of infinity” in the standard English sense of the word, but talking about ‘greater cardinality’ in the sense of the non-existence of an injective function from one infinite set to another seems sound.

It follows, alright, if you accept the concept of a set of infinite elements (i.e., an ACTUAL infinity).

25

It follows, alright, if you accept the concept of a set of infinite elements (i.e., an ACTUAL infinity).

I do not know why you say that, Tom. Why would a mathematical infinity require an actual infinity to be sensible?

26

It follows, alright, if you accept the concept of a set of infinite elements (i.e., an ACTUAL infinity).

It’s not so much something you ‘accept’ as something that’s demonstratably true. The set of integers has infinite elements. There’s no way you could even possibly begin to argue that it’s finite unless youre postulating some kind of intuitionism and the reconstruction of set theory.

edit: actually on second thoughts, intuitionism could possibly be the conventional philosophy of mathematics closest to what I think Rand would believe, based on IOE. I remember reading an article by David Ross where he went near this without stating it outright. Hmmmm.

27

… actually on second thoughts, intuitionism could possibly be the conventional philosophy of mathematics closest to what I think Rand would believe, based on IOE. I remember reading an article by David Ross where he went near this without stating it outright. Hmmmm.

Hardly. It is true that Ross atempts to connect intuitionism ala the likes of Brouwer to Objectivism via infinity and logic, but the connection between that and the actual ideas expressed by Ayn Rand are coincidental, not fundamental, and mostly irrelevant. Rejection of Cantor is not sufficient, and the intuitionist’s disregard for the law of excluded middle is not exactly a plus.

28

I do not know why you say that, Tom. Why would a mathematical infinity require an actual infinity to be sensible?

Well, for starters I will admit that I am not that well-versed in set theory (I only know the basic notation and concepts). However, I have had a year of (introductory) calculus, and have used the concept of infinity only with respect to limits. And in that case, I was taught to regard infinity only as an idefinite process, never as a cardinal number.

I’ll read some more into this; though I must admit I can only stand reading a book on axiomatic set theory for a few minutes before I have to put it down because my head just hurts. It’s so formal and abstract!

What do you think about axiomatic set theory are as a foundation for mathematics? It seems too formal and rationalistic to me, but…

29

Well, for starters I will admit that I am not that well-versed in set theory (I only know the basic notation and concepts). However, I have had a year of (introductory) calculus, and have used the concept of infinity only with respect to limits. And in that case, I was taught to regard infinity only as an idefinite process, never as a cardinal number.

But it is still a mathematical abstraction, not a physical existent.

I’ll read some more into this; though I must admit I can only stand reading a book on axiomatic set theory for a few minutes before I have to put it down because my head just hurts.  It’s so formal and abstract!

Well, like most learning you have to be motivated, and without a specific purpose in mind, complex mathematical theories are too abstract for casual reading.

What do you think about axiomatic set theory are as a foundation for mathematics?  It seems too formal and rationalistic to me, but…

I am not a big fan of set theory, but it is a perfectly valid and useful aspect of mathematics. However, I think that proper foundational issues in mathematics require a different basis than standard set theory. (Note also that there are several different versions of set theory.)

30

Although I am also not an expert, I share Tom’s reservations about set theory. The problem is that the concept “set” connotes a definite, finite group of objects, and even if some of the better mathematicians explicitly state that they are using it to denote something different (and as far as I know they don’t), it’s confusing to use it the way they do. I think “series” or something along those lines would be a much better term to describe open-ended “groupings” such as integers.

In my experience (in other words, at least in college classrooms), the bad terminology does in fact lead to significant philosophical errors on the part of many or most logicians and mathematicians, who often do at least implictly use the concept of “infinity” as though it is a quantity rather than a concept of method.

31

Although I am also not an expert, I share Tom’s reservations about set theory.  The problem is that the concept “set” connotes a definite, finite group of objects, and even if some of the better mathematicians explicitly state that they are using it to denote something different (and as far as I know they don’t), it’s confusing to use it the way they do.  I think “series” or something along those lines would be a much better term to describe open-ended “groupings” such as integers.

In order to properly criticize the use of terminology in a field you first need to understand the precise meaning of those terms in that field. But, even more importantly, you cannot condemn a theory because of the words used to describe the theory. One may argue that the use of the word “charm” to denote a particular property of a quark is an unfortunate choice of words, but to condemn the quantum theory of quarks based on the use of the word “charm,” would be improper.

Likewise for set theory. If you object to the process of adding unlimited elements to a set, then you must also object to the process of adding numbers in an unlimited fashion. Sets are mathematical abstractions, not physical entities, and they certainly are not necessarily defined in the field of mathematics as “connot[ing] a definite, finite group of objects.”

As Ash says in his post, I agree that bad terminology can lead to philosophical errors. But when concepts are clearly defined, that act, in and of itself, delimits the possibility of such an error. Mathematics is a very precise science, and its defintions are exemplary. Those not familiar with the field may not know those mathematical definitions, but the common sense meaning of a term is hardly reason to give one cause to reject an entire theory.

Note that in Objectivist epistemology a word is a symbol that denotes a concept, and that concept itself stands for an unlimited number of concretes, without implying that such an unlimited number of concretes necessarily exist in physical reality. Mathematical abstractions can be rather far removed from the chain that eventually leads to perceptual concretes, but we should applaud that level of abstraction, and not condemn it because it does not relate to some common sensical use of a term.

With that said, as I mentioned previously, I am not a fan of set theory per se, but I do defend it as a perfectly valid mathematical abstraction.

32

Those not familiar with the field may not know those mathematical definitions, but the common sense meaning of a term is hardly reason to give one cause to reject an entire theory.

I don’t recall saying that I rejected the entire theory on the basis of the one small criticism I made. I’m just curious why they would choose to use one word in a sense opposed to its common usage (possibly leading to confusion) when there are plenty of other words they could have easily used instead that would have been much closer in their common sense to the mathematical usage. It seems counterintuitive and counterproductive, and it makes me wonder if the abstraction was properly conceptualized by those who coined the term for it. Of course, it’s also possible that there are historical or etymological reasons for the choice of that term, I don’t know. But I really don’t care enough about this topic to say anything more.

I guess it may not be the theory I have a problem with so much as the way that it is taught (or at least the way it was taught to me).

33

There was a famous mathematician who wondered whether better sense could be made with set theory if there were limitations placed upon the “X is an element of Y such that …” formulations in defining sets (I’m not at home now, but it was in the beginning of A. P. Morse’s “A Theory of Sets.”)

I’ve come to the tentative conclusion that that those “limitations” for defining sets should be exactly what Ayn Rand did for regular concept formation, incorporating the ideas of Conceptual Common Denominator, distinguishing characteristic and measurement-omission. In other words, we need to create a foundation for sets which is grounded in perception, via the method of integration-differentiation.

As it stands, any type of willy-nilly mathematical expression can be used to form sets, and this leads to arbitrariness in the extreme, to the point that math becomes a “science of the arbitrary,” so to speak (which of course is an enormous contradiction.)

34

What do you think about axiomatic set theory are as a foundation for mathematics?  It seems too formal and rationalistic to me, but…

The problem is that the primitive terms are left “undefined.” In other words, such attempts are not grounded in perceptions, and are thus not proper methods of validating mathematics.

35

There was a famous mathematician who wondered whether better sense could be made with set theory if there were limitations placed upon the “X is an element of Y such that …” formulations in defining sets (I’m not at home now, but it was in the beginning of A. P. Morse’s “A Theory of Sets.”)

You are probably thinking of John Kelley. In the literature this is referred to as the MK set theory, for Morse-Kelley. But, I fail to see the relevance of even mentioning this.

I’ve come to the tentative conclusion that that those “limitations” for defining sets should be exactly what Ayn Rand did for regular concept formation, incorporating  the ideas of Conceptual Common Denominator, distinguishing characteristic and measurement-omission.  In other words, we need to create a foundation for sets which is grounded in perception, via the method of integration-differentiation.

I understand all of these words, individually, but I cannot make sense of what you are proposing, taking the words as a whole. Incidentally, for whatever it is worth, I personally think it more important to establish a foundation for mathematics independent of set theory, rather than “create a foundation for sets.” But, take that as an opinion since I am not in a position to establish that here.

As it stands, any type of willy-nilly mathematical expression can be used to form sets, and this leads to arbitrariness in the extreme, to the point that math becomes a “science of the arbitrary,” so to speak (which of course is an enormous contradiction.)

Sorry, but current mathematics is anything but a “science of the arbitrary,” as witnessed by the endless series of incredible mathematical accomplishments in the past century. Nevertheless, as I have always said, foundational mathematics is in its infancy, and needs a separate course from set theory.

36

I personally think it more important to establish a foundation for mathematics independent of set theory, rather than “create a foundation for sets.”

Yes, I agree. What I am saying about sets is that if we want to use the language of sets to serve as a foundation for math, then we need to establish the proper rules for forming sets. If you look at the implicit or “set builder” notation method of forming sets, you will see that a process of identifying a superset followed by a process of selecting a subset from within that superset is involved. This is suggestive of AR’s integration-differentiation method. My hypothesis is that we can use AR’s exact same method in building sets.

If we can, this then represents the death of set theory and the birth of a new mathematical epistemology. Concepts of mathematics will be shown to be formed by the same method of forming regular, nonnumerical concepts. That is my goal.

Sorry, but current mathematics is anything but a “science of the arbitrary,” as witnessed by the endless series of incredible mathematical accomplishments in the past century. Nevertheless, as I have always said, foundational mathematics is in its infancy, and needs a separate course from set theory. 

I wasn’t saying that all of math is a science of the arbitrary, since it is not monolithic. Some mathematicians see it that way, philosophers of math primarily. They say that math is “dense” and that humans can only get glimpses of mathematical truth or snatches of insights here and there. They confuse symbolism with meaning and think it makes sense to construct systems comprised of denotationless symbols. They say that all mathematical concepts are sets, and that “any mathematical expression” can be used in the process of forming a set. This is circular, since mathematical expressions were what they were supposed to be defining in the first place (i.e., they need to define what “set” means without using the concept of “set” in the definition.)

Certainly I agree that a lot of math is wonderful and we have done wonderful things with it, however, I do not believe that concepts of mathematics have been philosophically validated, meaning, no one has shown how all mathematical concepts properly reduce back to perceptions and reality. The situation is analogous to the state of philosophy before AR. Human concept formation was not philosophically validated, though humans had used concepts to do great things before she validated it.

37

… They say that math is “dense” and that humans can only get glimpses of mathematical truth or snatches of insights here and there. They confuse symbolism with meaning and think it makes sense to construct systems comprised of denotationless symbols.

I do not know you – neither your background nor your knowledge – so this is not addressed directly to you. Nevertheless, I have found, over the years, those most vocally antagonistic to current mathematics, and, physics too for that matter, are typically those who do not understand the math, or the physics. I do not defend the platonism in mathematical theory, and I do not defend the rationalism in physics, but both of these structures are incredibly marvelous achievements that cannot really be appreciated by those who are ignorant of the language.

It seems easy for some to sit back and berate the mathematicians for using symbols and complex structures that requires a great deal of knowledge to understand. The more I continue to study and learn the more I appreciate the brilliant minds that have touched both of these fields. There is much to fault in currrent mathematics and physics, but few of the critics that I hear – and, I am sad to say, this is particularly true among Objectivist critics – know the subject that they criticize. Philosophy is a powerful tool, and Objectivist philosophy is the most powerful of them all, but philosophical principles can only guide your knowledge when it comes to science. Criticism based on ignorance is worse than no criticism at all.

38

I’m very sympathetic to what you are saying. I too admire what I consider virtuoso performances on the part of certain mathematicians (for example, Galois having created group theory.) Such is possible even without mathematics having been explicitly validated. I would compare it to a gifted writer composing an epic novel, while not explicitly understanding all the grammatical-epistemological nuances of his language as a grammarian would.

39

Regarding the essay by Alex S., linked to by Stephen Speicher, above, I had a minor problem/question with:

“there is nothing in the Law of Identity that mandates every existent possess a (finite) size, anymore than the Law of Identity mandates every existent be (finite) in time. The concepts of size and time apply to certain existents, which have specific natures that allow for such applicability.”

Is it strictly a scientific issue to insist that every existent possess a finite size and time? Or is that near the edges of what philosophy can say on the subject? It seems rationalistic to claim, on one hand, that existence necessarily implies identity, but that the identity itself need need not consist of anything we really know for sure – maybe size, maybe time… but maybe something else entirely.

Can’t one say at least that size and time are necessary qualities of existents? Even if future knowledge should amend our contextual understanding of these concepts, the evidence for the existence of these qualities, at least, seems universal.

Another question I have with the above quote is that it is used specifically in the essay to refer to the universe as a whole. As such, it is a confusing illustration, because the object of its point, is the only exception to the illustration; it says, in effect: “that the universe has no size or time does not violate the law of identity, since the law of identity does not stipulate time or size as a necessary attribute of reality, even though every existent *does* have time and size… except the universe”.

40

Do you have any specific objections to Cantor’s work, in the sense of flaws that you can find in it? The idea of ‘different levels’ of infinity follows logically from the definition of sets, functions and cardinality; its not really something that you can argue with, unless you have a problem with those definitons.

Don’t make the mistake of buying into Rationalist premises. Take for example, the linguist who subscribes to the premises of “generative grammar.” He would think it makes sense to talk about “blackbirds” if one only has the concepts “black” and “bird” already formed and then “puts them together.”

But it doesn’t make sense. In order for the term “blackbird” to have real meaning, one would have to know that there is such a thing. You can’t just take concepts and combine them in the absense of knowledge about the real world and claim that you have knowledge.

The same thing goes with Cantor and set theory. You can’t take sets or properties of sets and then make identifications, which, in order to be true, would need to be shown to be so using induction, but then try to stick to using only deductive means. This is primacy of deduction and is Rationalistic.

In an inductive process (not to be confused with the misnamed process of “mathematical induction,” which is not inductive at all, but is deductive)–in a truly inductive process, one needs to make reference to reality every step of the way. This is exactly what Cantor does *not* do when he Rationalistically makes reference to power sets without assigning them any real-world referents. It’s like the “generative grammarian” who thinks he can crank up his linguistic machine (his generative “grammar”) and produce meaningful sentences.

That being said, if there is any sense to made out of transfinite numbers, such truths are only implied by Cantor’s theory. One would have to take his theory apart and reconstruct it in a proper manner (which would be analagous to the generative grammarian who deals with real linguistic phenomena, but doesn’t know how to represent such truths in a proper theory. He’s only able to toss things around as floating abstractions.)