Physics Lournal

The Study of Mathematics

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Author:: Bertrand Russell

Tags:: Mathematics Philosophy

Bertrand starts this essay by admitting that, every human activity, rightfully, should be questioned with regards to its purpose, and how it "contributes to the beauty of human existence?"

Even the pursuits that due nothing but continue life, should cause one to bear in mind that it is not mere life that is to be sought after, but life living in the "contemplation of great things.

Particularly, with endeavors that have no end aside from their own pursuit, we should particularly strive to keep alive a knowledge of the goals of these endeavors.

In his eyes, however, this responsibility is not being handled by those who would be tasked with doing so. In fact, this duty has been so thoroughly neglected, that to posit a question about the value of an endeavor or field of study, is considered uncouth.

He feels as if, more than anything, the study of Mathematics, has suffered from this degree of negligence.

While it is still expected to be understood at a fundamental level, the reasons for the importance of the study, are not imparted along with the fundamental knowledge.

Generally, if asked what value the study of Mathematics provides society, people will point to the economic and utilitarian benefits of it.

Alternatively, some will reply that the study of the field will train the reasoning faculties, however, these same people will remain attached to and supportive of the logical fallacies their faculties of reason should alert them to the existence of.

However, in the eyes of Bertrand, neither the training of reason, or utilitarian benefits are the reasons that we should study Mathematics.

Plato seems to have felt that contemplating Mathematical truths was an activity that was worthy of the Deity.

Mathematics, Bertrand states, possesses not only a truth, but a supreme beauty, cold and austere, without any temptation provided for our more sensuous and base appetites. More importantly, it offers the opportunity for "..the sense of being more than man...", the touchstone of the highest excellence.

This to me clearly seems to be echoing Plato's ideal world of forms.

The content of mathematics, should not just be learned as a tool, but should be incorporated into every day thought. The reason for this is that, for most of every day life, we find ourselves forced to compromise between the ideal, and the possible, but the world of pure reason (the platonic world, perhaps), offers, nor knows of reason, compromise or limitation.

Reason is generally put forth as the demonstration of the truth of the obvious, but it can also be a demonstration of non-obvious truth.

Unaffected by (human) passion, the "pitiful facts of nature", mathematicians have assembled an "ordered cosmos", where pure thought can dwell, and our "nobler impulses" can escape the "dreary exile of the real world".

I cannot help but contest the idea that our cosmos is properly ordered and believe that what Bertrand appreciates about mathematics, is that it is a cosmos created by order in terms of human understanding, which reminds me of the feynman quote about the absurdity of nature:

Quantum mechanics describes nature as absurd from the point of view of common sense. And yet it fully agrees with experiment. So I hope you can accept nature as She is - absurd.

Even the gifted children, in the beginning of algebra, experience difficulty: the use of letters is a novel experience which seems to have no purpose but to obscure.

One might assume that each letter stands for some particular value, but the fact is that algebra is the first introduction of general truth, which do not hold for just for a specific thing, but for any, perhaps even an infinite number of things.

It is the ability to understand these more general truths that gives the intellect power over the world of "things actual and possible", and the ability to grapple with this generality is something a mathematical education should confer to a student.

However, it is more likely, that the teacher of algebra cannot explain precisely what separates it from arithmetic, and is thus unable to assist the student with gaining comprehension.

What instead takes place is that, just as with arithmetical education, rules are presented to the student, without explanation of their usefulness, and the student undergoes rote memorization, and attains a false sense of mastery of the concept as a result of being able to find an answer to a question, which does not imply inner comprehension of the nature of the process used to find the answer.

Once algebra has been learned, the student progresses smoothly until they have to grapple with the concept of infinity, such as in the infinitesimal calculus, and higher mathematics.

The creation of solutions of the difficulties which surround the concept of the infinite, may be the greatest achievement of the current mathematical age, providing an answer to Zeno's paradox, found by Cantor and Dedekind: prior to their work it was assumed that if, from any collection of things, some items were removed, the resulting total must be less than the original number of things.

This assumption only holds for finite collections, and is rejected by the infinite, thus removing many of the things that harangued human reason, allowing for an exact science of the infinite.

{{TODO}} Finish reading this essay. {{∆:1+0}}

Referenced in

Why Mathematics?