A few days ago, Xamuel asked if mathematics was subjective or objective. After discussing the question, he came to this conclusion:

One could make the program more sophisticated, and have it make definitions in such a way as to ultimately minimize the sheer complexity of its theorems. But could it come up with those names in a good way? Going about it the most obvious way, it might define a noun “N9524″ to mean what you and I mean by “prime number”, simply because that happens to be the 9,524th noun that it defines (and in reality, 9,524 is probably a hilarious underestimate of a cosmically infinitesimal nature). That’s assuming its definitions match up with ours at all. More likely, its definitions would have only a loose relationship, if any, to human-made definitions. To get around this, to give these nouns halfway decent names, would probably require full-on, post-singularity, open-the-pod-bay-doors-Hal, Artificial Intelligence. (Or maybe I’m wrong. Many a pessimistic futurist has been humiliated in the past…)

So to sum it all up? Mathematics must necessarily be somewhat subjective as a defense mechanism against trolls who could otherwise flood the journals with true-but-useless flotsam and jetsam. But it is far more objective than any other discipline in the world.

I’m inclined to disagree with his analysis, but only because I approach the question differently. Xamuel’s answer, assuming I understand him correctly, is that determining mathematical truths must be an inherently subjective activity because humans must determine what truths are correct

*and*relevant. Some assertions may be incorrect; others may be correct but either irrelevant or redundant. Humans are therefore needed to sort out useful axioms from unnecessary or wrong axioms.Xamuel’s answer, then, is quite correct in that it recognizes the inherent human element in the discipline of mathematics. Where he and I differ, at least insofar as I can divine his beliefs via his public writing, is that I would assert that mathematics, as a discipline, is fundamentally subjective.

I say that mathematics, and mathematical truths, are fundamentally subjective because the system is inherently axiomatic. The number 2, for example, is self-defined. The statement “1 + 1 = 2,” though quite correct, is also quite subjective. The numbers and relational symbols are inherently axiomatic and, as such, the truth of the statement is inherently axiomatic. Since all other mathematical truths are either axiomatic or based on reductive axioms, it stands to reason that the entire discipline is fundamentally subjective in nature.

Of course, this doesn’t mean that mathematics is without value. In fact, its value is derived from its wide-ranging application. Businesses make use of mathematics every day, as do virtually all scientists. Mathematics’ value is that it is perfectly logical, consistent in its relationships, and very abstract.

Mathematics’ high degree of abstraction is, in my opinion, its greatest value. You can have a symbol or two stand in place of large amounts of data and then derive relationships between various data. The ubiquity of the Cartesian plane should speak to the value of this highly abstract method of analysis.

Additionally, the high degree of abstraction allows mathematics to be treated as an objective method of analysis because it is less subject to the vagaries of human perception, which is, I believe, the point that Xamuel was trying to make. There are very few fundamental shifts in mathematical understanding. The most recent, as I define the phrase "fundamental," was the shift from Roman numerals to Arabic numerals, which applied to mostly western mathematicians, and the invention of “0.” Pretty much everything else since then has been built upon this system.

*Pi*, for example, is commonly understood within the framework of Arabic numerals (i.e. 3.14159…).As it stands, the fundamental axioms of mathematics (the base-ten Arabic numbers, e.g.) are so well-established that it is easy to treat them abstractly and relate them to real world applications. So, even though “2 + 2 = 4” is inherently subjective, it can be treated as an objective truth because no one disagrees with it. Thus, mathematics is considered objective because, in part, there is no point in disagreeing with its inherent subjectivity.

(For further reading, see “The Limits of Science,” from my book

*The Early Years*. Also see Xamuel's post on 1 + 1.)
That is a good point... a priori, it is a little subjective which axiom systems to use. Sometimes, we have a situation where a bunch of differing axiom systems can all be proven to be equivalent (as is the case in the various different axiomatizations of set theory), and that lends some objectivity to them. And with "1+1=2", this is true by definition, but why do we use the symbols "2" and "3" for 1+1 and 1+1+1 respectively... as symbols, they are fairly arbitrary (though I suppose in the case of 3, linguists would probably argue it gradually evolved from three horizontal lines stacked together)

ReplyDeleteBy the way, you're a mind reader. I was actually planning on writing an article about "1+1=2" :)

@Xamuel- of course, 1+1 doesn't always equal 2. If I remember correctly, binary logic states that 1+1=0. Of course, this assertion is inherently axiomatic as well, and the binary system is completely logical and consistent, and has its own application. At any rate, its always useful and occasionally fun to dwell on the inherent subjectivity of human perception.

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