[Avodah] O(n^2) algorithm for matrix multiplication

[Akiva Miller]

R' David Cohen wrote:
> Yishuvo shel olam aside, I also see inherent value in
> learning math: it is truth.  We learn Torah because it is
> revealed truth, and some of us study science because it is
> another reflection of the ways of Hashem, who is truth. The
> laws of math are objectively true, independent of their
> mapping onto the physical world.

R' Daniel Israel added:
> Certainly this would then apply to the sciences ... I'm
> curious if you would say this about humanities. The truths
> they expose are of an altogether different type.  And art,
> which doesn't generate truth, does help "fill out" all the
> potentialities of HaShem's creation.

In strong support of both of these views, I cannot help but quote the words of Rav Joseph B Soloveitchik in "The Lonely Man Of Faith". (For those who are unfamiliar with this work, I will give a too-brief introduction to two terms he uses throughout it: "Adam the first" and "Adam the second" -- mankind as described by Bereishis 1 and Bereishis 2 respectively.)

--- the last three paragraphs of Chapter 1 begin here ---

Hence, Adam the first is aggressive, bold, and victory-minded. His motto is success, triumph over the cosmic forces. He engages in creative work, trying to imitate his Maker (imitatio Dei). The most characteristic representative of Adam the first is the mathematical scientist who whisks us away from the array of tangible things, from color and sound, from heat, touch, and smell which are the only phenomena accessible to our senses, into a formal relational world of thought constructs, the product of his "arbitrary" postulating and spontaneous positing and deducing. This world, woven out of human thought processes, functions with amazing precision and runs parallel to the workings of the real multifarious world of our senses. The modern scientist does not try to explain nature. He only duplicates it. In his full resplendent glory as a creative agent of God, he constructs his own world and in mysterious fashion succeeds in controlling his environment through manipulating his own mathematical constructs and creations.

Adam the first is not only a creative theoretician. He is also a creative aesthete. He fashions ideas with his mind, and beauty with his heart. He enjoys both his intellectual and his aesthetic creativity and takes pride in it. He also displays creativity in the world of the norm: he legislates for himself norms and laws because a dignified existence is an orderly one. Anarchy and dignity are mutually exclusive. He is this-worldly-minded, finitude-oriented, beauty-centered. Adam the first is always an aesthete, whether engaged in an intellectual or in an ethical performance. His conscience is energized not by the idea of the good, but by that of the beautiful. His mind is questing not for the true, but for the pleasant and functional, which are rooted in the aesthetical, not the noetic-ethical, sphere.

In doing all this, Adam the first is trying to carry out the mandate entrusted to him by his Maker who, at dawn of the sixth mysterious day of creation, addressed Himself to man and summoned him to "fill the earth and subdue it." It is God who decreed that the story of Adam the first be the great saga of freedom of man-slave who gradually transforms himself into man-master. While pursuing this goal, driven by an urge which he cannot but obey, Adam the first transcends the limits of the reasonable and probable and ventures into the open spaces of a boundless universe. Even this longing for vastness, no matter how adventurous and fantastic, is legitimate. Man reaching for the distant stars is acting in harmony with his nature which was created, willed, and directed by his Maker. It is a manifestation of obedience to rather than rebellion against God. Thus, in sum, we have obtained the following triple equation: humanity = dignity = responsibility = majesty. 

--- end of excerpt from "The Lonely Man Of Faith" ---

It seems clear to me that Rav Soloveitchik puts great value in our study of the sciences, and even of the arts, as RDC and RDI wrote above. The part that is not so clear to me is *how much* value it has, and the *nature* of that value.

Specifically, I think that the Rav could be understood as simply describing human nature, that is, the drives which we innately have, and which were put into us as part of Creation. But it could be more: When we exercise those drives, we fulfill the command to (as he translated it) "fill the earth and subdue it."

If so, then the search for an "O(n^2) algorithm for matrix multiplication" is not merely an enjoyable and stimulating pastime, but it is (or can be) a fulfillment of our role in creation. My remaining questions are:

- whether "our role" in this context refers to humanity in general or Jews in particular,
- whether this fulfillment is of a chiyuv or a mitzvah,
- whether one has accomplished this even when done for personal reasons, or whether one needs to have higher kavanos in mind.

Akiva Miller

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