This is very interesting, but it is not clear to me how mathematical truths being objective resolves the intrinsic vs. subjective dispute. What is the nature of the difference between intrinsic and objective?

Whitehead argued that mathematical truths are contextual, but objective within a specified context. He liked to use a simple example: If you take 2 apples and add 2 apples you have four apples. However, if you take two raindrops, and add them to two additional raindrops, you get one big raindrop. His point, I think, was that even a relatively simple mathematical procedure like addition has assumptions which make the operation viable. In this case the assumption is a certain level of discreteness; that is one can count the individual apple units, but the discreteness of raindrops is not so clear. However, Whitehead did not think of mathematical truths as cultural artefacts or subjective whims. Is this kind of analysis leaning toward the objective as indicated in your post?

Thanks,

Jim 1209297638:::Burgess Laughlin:::burgesslaughlin@macforcego.com:::http://www.aristotleadventure.blogspot.com:::Jimfw: "Whitehead argued that mathematical truths are contextual, but objective within a specified context."

Jim, I am not a mathematician. However, for other reasons, I am interested in the various meanings of "objective" in the history of philosophy.

What did Whitehead mean by "objective," if that is a term he himself used? 1209300737:::Alfred Centauri:::alfredcentauri@bellsouth.net::::::"If you take 2 apples and add 2 apples you have four apples. However, if you take two raindrops, and add them to two additional raindrops, you get one big raindrop. His point..."

is that a raindrop is measure of an _amount_ of water and thus measures add just as surely as apples do. 1209307089:::Nate T.:::::::::As an aspiring mathematician whose trade is dealing in concepts, this topic interests me greatly, and I've often thought of how Objectivist principles would apply to the philosophy of mathematics.

As I understand it, much of the work in the philosophy of mathematics is an attempt to resolve the issue of the axiom of choice and other such "nonconstructive" postulates. I'm not so much bothered by other philosopher's confusion about the nature of concepts (Rand cleared that part up for me) as I am by this issue (since the axiom of choice is used in my studies from time to time).

Paul, do you know whether Dr. Corvini (or any other Objectivist scholar) has done any work on answering this question? From your post, it seems that she mostly focuses on the role of numbers in mathematics in her courses. 1209315932:::Jim May:::seerak@gmail.com::::::I too have had a lot of interest in this topic -- unfortunately, I lack the theoretical background to do anything more than speculate about such matters (I studied applied mathematics in an electronics engineering context). I'm glad to see that someone with that background is digging into this sort of thing. 1209321992:::Adrian Hester:::::::::Alfred Centauri: "a raindrop is measure of an _amount_ of water and thus measures add just as surely as apples do."

Well, that depends on whether you mean weight or volume; volumes do not add in the same way in general. If you mix two volumes of liquid, the volume of their mixture is not in general the sum of the volumes added in general; only if they're the same liquid (either a pure compound or the same solution at the same concentration) at the same temperature (or if you're very lucky under other circumstances). This is quite different from adding an apple and an orange, in which case you always have two pieces of fruit. (But assuming there are no chemical reactions causing a gas to be produced or nuclear reactions, heh, then the mass of the mixture *is* always the sum of the masses added.) 1209326530:::Vic P:::::::::The human mind has built-in mathematical functionality i.e. the parallax performed by our ocular muscles in our visual system when we perform depth perception. Staring at the starry sky at night produces a cosmic feeling in us because our mind calculates distance subconsciously.

1) Line up 10 matchsticks or 10 toothpicks or 10 of anything on a table.

2) Count the objects to yourself: "one..two..three..four..five..six..seven..eight..nine..ten"

3) Now count the objects again except concentrate on counting them in the left lobe only of your brain: "one..two..three..four..five..six..seven..eight..nine..ten"

4) What's the difference between steps 2 and 3?...In step 2 you were counting objects or performing arithmetic while in step 3 you were assigning numerical identity or performing mathematics.

5) When you perform arithmetic you are using both brain lobes or making conscious identity with reality.

6) When we perform arithmetic we essentially perform comparitive identity by assigning the odd numerical items to our left lobe and the alternate even item to the right lobe. When we count objects or perform arithmetic we essentially count in 'number sets': "one (left lobe)..two (right lobe)..(set 1)...three (left lobe)...four (right lobe)...(set 2) etc."

7) When we perform arithmetic counting our nervous systems act like a zipper because we essentially see or create internalized distinct objects or object sets as memory in our subconscious.

8) Set theory of course is based in arithemetic principles and the principle of internalized identity of REAL or IMAGINED OBJECTS.

9) Most human beings like to employ symetrical thought and perform arithmetic in even numbers. If we state an amount of "1" ONLY we are also implying the absence of "2" or completion of the first subset of integers.

10) Because we see numbers in pairs, the number of even numbers is equal to the total number of odd numbers and they are both equal to the number of total numbers since the number of total numbers equals the number of number pairs or 'number sets'.

11) We build 'number sets' into higher sets and the first set of higher sets occur because our minds control 5 fingers on each hand or 5 sets of 2 = 10

12) In step 3 when you were counting the objects in your left brain lobe only, you were not really counting the object but essentially identifying it's position in space as we would identify points on a line.

13) Perform the counting in the right brain lobe and you have identified points on an imaginary spatial line or the line of imaginary numbers. If you mentally rotate the axis of the imaginary numbers in the right hemisphere and project the axis of real numbers from the left to the right hemisphere you create the 2 dimensional axis of real and imaginary numbers. Likewise we can create the third axis as well in our right hemisphere of spatial reasoning.

14) When we perform pure mathematics we are not perceiving objects in our subconscious but operating on our spatial reasoning in our right brain lobe with logical words (numbers) from our left brain lobe. We are employing laws or rules or of course trained reason stored in our subconscious memory. 1209335158:::Paul Hsieh:::paul(at)geekpress(dot)com:::http://www.geekpress.com:::NateT: My understanding is that Pat Corvini is working on a book on the philosophical foundations of math from an Objectivist perspective, and that her OCON courses are drawn from that material. I don't know if she is going to specifically cover the Axiom of Choice, but I think it will cover aspects of set theory. I'll try to ask her when I attend her 2008 course, as well as see if there will be an ETA on her book. 1209335370:::Paul Hsieh:::paul(at)geekpress(dot)com:::http://www.geekpress.com:::Jimfw: Your question is a good one, and it's part of the broader distinction that Objectivists draw between "subjective", "intrinsic", and "objective" concepts in general. It's covered in Ayn Rand's book, "Introduction to Objectivist Epistemology" and Leonard Peikoff's book, "Objectivism: The Philosophy of Ayn Rand".

Unfortunately, I don't have time tonight to sketch out a summary of the theory. But perhaps another commenter will be willing to step up. Otherwise, I'll see if I can post something in the next few days. 1209367581:::Jimfw:::jimfw@hotmail.com::::::Dear Burgess:

Whitehead discusses objectivity at great length in his works. Whitehead is a difficult philosopher because he often coined his own terminology. As a mathematician he was attempting to be precise where previous philosophers had been muddled, particular with regards to the nature of identity and process. Thus he felt a strong need to invent new words in order to not mislead his readers and clarify his arguments. But this does make his works hard to enter into, though in my opinion it is worth the effort.

Having said the above, here are two references regarding objectivity from "Process and Reality:

The Category of Objective Identity: There can be no duplication of any element in the objective datum of the 'satifaction' of an actual entity, so far as concerns the function of that element in the 'satisfaction'.

The Category of Objective Diversity: There can be no 'coalescence' of diverse elements in the objective datum of an actual entity, so far as concerns the functions of those elements in that satisfaction.

'Coalescence' here means the notion of diverse elements exercising an absolute identity of function, devoid of contrasts inherent in their diversities.

(Process and Reality, page 26)

Note: 'Satisfaction' is not a psychological category in Process and Reality but refers to the prehension of becoming by actual occasions as uniquely determined by each actual occasion or entity. That is to say, each actual occasion determines its prehension of becoming to its own satisfaction.

Later Whitehead write the following about objectivity:

There are three main categoreal conditions which flow from the final nature of things. These three conditions are (i) the Category of Subjective Unity, (ii) the Category of Objective Identity, and (iii) the Category of Objective Diversity. Later we shall isolate five other categoreal conditions. But the three conditions mentioned above have an air of ultimate metaphysical generality.

The first category has to do with self-realization. Self-realization is the ultimate fact of facts. An actuality is self-realizing, and whatever is self-realizing is an actuality. An actual entity is at once the subject of self-realization, and the superject which is self-realized.

The second and third categories have to do with objective determination. All entities, including even other actual entities, enter into the self-realization of an actuality in the capacity of determnants of the definiteness of that actuality. By reason of this objective functioning of entities there is truth and falsehood.

(Pages 222-223)

If one refers to the index of Process and Reality the references to objectivity are very numerous; it is one of Whitehead's main concerns. The above quotes just touch what he has to say.

Best wishes,

Jim 1209472853:::Mike Hardy:::hardy@math.umn.edu::::::I'm inclined to think Paul Hsieh is right to say the solution to at least some of these philosophical issues lies in discarding both the subjectivist and intrinsicist positions. But I don't think that gets at the issue of constructive versus non-constructive arguments, at least until you say a lot more than that. As for the issue of the Axiom of Choice: it seems to be almost a cliche to say that that is non-constructive, but that in itself is certainly a debatable point. Look at Erret Bishop's take on it in his book on constructive analysis.

For a much simpler example of what the constructive-versus-non-constructive issue is about, consider this. "Goldbach's Conjecture" says ever even number greater than 2 is the sum of two prime numbers. For example

100 = 3 + 97

or

100 = 11 + 89.

etc. No one has found a counterexample, nor proved that this pattern must continue forever. So now suppose someone comes along and deduces a logical contradiction from Goldbach's conjecture. Then says "Since Goldbach's Conjecture is therefore proved false, there must exist some even number greater than 2 that is _not_ the sum of two primes." That would be a "non-constructive" proof of the existence of such an even number; it gives no way to actually find that number.

Some people reject non-constructive proofs on more-or-less philosophical grounds (Erret Bishop, mentioned above, was one of those), in some cases saying they amount to applying the Law of Excluded Middle to things that don't really exist. (For an uncontroversial example of applying the Law of Excluded Middle to things that don't really exist, consider an example: _The_Fountainhead_ tells a story that begins in the spring of 1922 and ends in the spring of 1940, about Howard Roark, who was born in either 1899 or 1900. Consider the statement: "Either Howard Roark died in 1986 or Howard Roark did not die in 1986." Is that a valid use of the L. of E.M., in view of the fact that Howard Roark never really existed?) 1210709372:::A-non-emous:::::::::Yes it is. Howard Roark does exist, as a fictional character as described in The Fountainhead. And the events in the story in which he is a part, or those described by the author, constitute his reality or existence. One cannot drop the context of the meaning of "his existence." The book created the conditions for Roark's existence.