Mathematics

What is the Objectivist view of mathematics? How does Objectivism define numbers? Does it use set theory, for example? Does anyone know?

I recall somewhere in IOE Ayn Rand says that the claim 2 + 2 = 4 is a necessary truth because 2 means one plus one and 4 means one plus one plus one plus one.

If that's what the numbers mean then, how would you prove that 2.1 + 1.9 = 4. After all, the definition of 4 is 1+1+1+1...

Any ideas?

(Just to let you know where I'm going with this, keep in mind that every number has an infinite number of independent relationships to every other number. For example, not only are there an infinite number of subtraction expressions that equal one, there are an infinite number of division statements, logs, lns, etc)

Post 1

Thursday, May 16, 2002 - 11:19pm

For one Objectivist's approach to mathematics, go to The Intellectual Activist (intellectualactivist.com). Go to the store, then to back issues, and do a search for "Pisaturo". You'll come up with a whole bunch of issues with his work in them. I think some of them are available online somewhere, too -- an altavista/google search for his name might turn something up.

Post 2

Thursday, May 16, 2002 - 11:47pm

Tried that...I just got excerpts that had polemics without a hint of explanation. I don't want to buy any issues unless I get a hint of what his theory is and whether it's plausible.

I hope he's not another one of these people that doesn't understand Godel and thus describes him as the Satan of mathematics.

I shall keep looking...

I thought Frege and Russell had promising theories of numbers, but the 20th century showed that set theory won't quite do it. Neither does category theory, apparently. And defining numbers as being numbers plus other numbers begs the question and presupposes an infinite set of axioms, as Godel pointed out.

Set theory, by the way, defines the number three as (for example):

There is some x such that Px and there is some y such that Py and there is some z such that Pz, and x is not y and y is not z and x is not z, and for all w such that Pw, w=x or w=y or w=z.

This is a way of counting (and thus defining a number) without presupposing numbers.

This is great, but when we start dealing with infinity...watch out!

Post 3

Thursday, July 4, 2002 - 1:00am

A theorem attributed to Ayn Rand on a particular webpage, is the following:

"In a triangle the inscribed circle touches the circle that passes through the midpoints of the sides."

I was wondering whether any soloists knew of an association of this theorem with Ayn Rand or of its occurrence in Objectivist literature. It would be interesting to know Rand's method of proof, or so I thought.

I've since realised, however, that the webpage in question ("Become an Objectivist in Ten Easy Steps") had me fooled. At first glance it seems like a promotion of Rand's ideas, but it's intended as parody. (Google describes it as such, and it's linked to from a page listing criticisms of Objectivism.) The more I look at the page now, the more I see that gives the game away.

Nevertheless, the theorem is true. Was the author of the webpage out to embarrass Rand on the grounds that she was beaten to the result by Feuerbach (1800-1834), or is it a lie that she discovered the theorem (independently)? Allowing for the slim possibility that the former is the case, I tentatively let my above enquiry stand.

Can anyone enlighten me?

Post 4

Wednesday, August 21, 2002 - 3:41am

Into the infinite comes the... Tautological???

Post 5

Wednesday, August 21, 2002 - 4:50am

I would say that numbers come from the concept of identity, and that the concept in mathematics come from concept-formation, just like all other concepts.

Apart from that, the particular theorems are the province of mathematics, not philosophy. Much like in science, philosophy provides a basis, but it doesn't dictate results. It's nature that decides.

Post 6

Monday, July 21, 2003 - 2:05pm

Objectivism has a view on mathematics? Explain, because I am no Objectivist expert.

Post 7

Monday, July 21, 2003 - 7:23pm

Eudaemonia was trying to figure out how Objectivism approached mathematics, Nate. AFAIK, Rand never dealt specifically with mathematics, as her education was primarily in history and philosophy. Rand probably wouldn't have had the context to specifically approach mathematics.

Post 8

Monday, July 21, 2003 - 8:10pm

Not so fast, Matthew

I seem to recall reading that Rand had a passion for mathematics and spent some time studying it for pleasure. Whether she wrote about it from a philosophical point of view I don't know.

No doubt other Rand scholars can chip in here and help us out.

Post 9

Tuesday, July 22, 2003 - 3:55am

Did she, Jonathan? I didn't know that. Where did you read that Rand was interested in maths?

Post 10

Tuesday, July 22, 2003 - 4:11am

It was in the following book which I read a month or so ago. Have returned it to the library now so cannot quote from it.

On Ayn Rand / Allan Gotthelf.
Published: Australia ; Belmont, CA : Wadsworth/Thomson Learning, c2000

Post 11

Tuesday, July 22, 2003 - 8:09am

Hmm. I had just finished reading Sciabarra's Ayn Rand: the Russian Radical last night, but I'm not done grokking it.

Post 12

Tuesday, July 22, 2003 - 5:11pm

Grokking??

Tuesday, July 22, 2003 - 5:54pm

I see you haven't read Heinlein. Get a copy of Stranger in a Strange Land.

Post 14

Saturday, December 6, 2003 - 4:01pm

In school, mathematics was always one of her favorite and best subjects. All through her life, Ayn Rand was fascinated by the connections between mathematics and epistemology. She said that it represented the abstract pattern of rational thought.

Near the end of her life, in her seventies, she began taking lessons in mathematics in order to investigate those links.

The original poster's summary does not sound like anything Ayn Rand would ever say. INTRODUCTION TO OBJECTIVIST EPISTEMOLOGY does not seem to contain any sentence resembling the one given. The reference to "necessary truths" sounds like it comes from someone who believes in the analytic-synthetic dichotomy, an error Rand specifically attacks, and which is discussed at length by Leonard Peikoff in that same book.

The question "How would you prove that 2.1 + 1.9 = 4? After all, the definition of 4 is 1 + 1 + 1 + 1" is based on a premise that lies at the root of that false dichotomy: the idea that a concept's referents consist of its definition. The truth is that a concept refers to an entity including all aspects not specified in the definition; the definition specifies the distinguishing characteristics only, those that explain the most about a concept, but the meaning of the concept itself includes everything.

In the case of a number, the meaning of 4 allows for every possible means of arriving at that digit via mathematics. By the way, that does not look like a definition Rand would subscribe to.

Post 15

Wednesday, January 21, 2004 - 8:34pm