The Larson Research Center

In the previous post, I wrote about Sir William Rowan Hamilton’s lamentation of the fact that algebra lacks an inductive basis.  There can be no satisfactory science of algebra, in his view, if it is to be based on the magnitudes of substance.  We see how he felt that while geometry is an inductive science of space, based on Euclidean geometry and the postulate of parallel lines, algebra is not, because negative and imaginary quantities, so essential to algebra, simply make no sense. He wrote:

…it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing.
 

Today, hardly anyone shares these misgivings publicly. We teach negative numbers in grade school (about the fifth grade these days) without fussing over the philosophical torture to young minds that we are inflicting, but still, no doubt, causing a few “crying jags” along the way.  Then we go on to teach kids imaginary numbers in middle school, blissfully unaware that we are teaching them a mathematical conundrum, where numbers denote magnitudes that “can neither be greater than nothing, nor less than nothing, nor even equal to nothing.”

Yet, we never teach why negative magnitudes are useful, even though they don’t make sense intuitively, nor why negative squares are useful, even though there is no escape from the conundrum they represent. Students are simply expected to learn the rules that are known to work, in spite of these philosophical contradictions, and they do learn them, for the most part, but at what cost?

Actually, even if we could measure and appreciate the cost, there’s not much that can be done about it, because no one really knows why these concepts of numbers are so useful as an art, and so consistent as a language, in spite of the complete lack of intuition concerning them.  Clearly, the only thing we could do is admit this failure to the rising generations,  but instead of doing that, we just don’t think about it any more, letting it sink deeper and deeper into obscurity. 

Well, at least we don’t think about it as Hamilton thought about it. He thought about it in terms of advancing science.  He wrote:

The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to light, and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful.

Of course, he articulated these thoughts, because he believed that there was a glimmer of hope that an inductive science of numbers could be developed from intuition, even though he was somewhat apologetic about it.  He refers to this as a “rudiment” of hope and asked indulgence from his contemporaries to explore it:

Indulgence, therefore, may be hoped for, by any one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a Science of Algebra: a Science properly so called: strict, pure and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning.

He called this “rudiment” of hope that he believed in the “Intuition of Time.”  He explains:

This belief involves the three following as components: First, that the notion of Time is connected with existing Algebra; Second, that this notion or intuition of Time may be unfolded into an independent Pure Science; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; the second of a deduction; the third is the joint result of the deductive and inductive processes.

Really? The notion of time is connected with existing algebra? Just try to find that conclusion in a modern textbook of algebra!  Yet, Hamilton understood that this is the essential difference between algebra and geometry:

The argument for the conclusion that the notion of Time is connected with existing Algebra, is an induction of the following kind. The History of Algebraic Science shows that the most remarkable discoveries in it have been made, either expressly through the medium of that notion of Time, or through the closely connected (and in some sort coincident) notion of Continuous Progression. It is the genius of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what it reasoned on as fixed.
…Lagrange considered Algebra to be the Science of Functions,* and…it is not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence as consisting in a Law connecting Change with Change. But where Change and Progression are, there is Time. The notion of Time is, therefore, inductively found to be connected with existing Algebra.

Thus, he begins to reason on the concept of a continuous progression in earnest, with the aim of developing an inductive science of algebra. He perceives that the continuous march of time, as counted by successive thoughts, if nothing else, means that there is past, present, and future: It means that counting moments of time implies that one moment is later, or earlier, than another, and that, taken together, this relationship is mathematical, because it leads to the following equations:

1) A = B,
2) B > A,
3) A < B,

where 1) can be compared to a point, and 2) and 3) to a line, or an interval, with “direction,” which he calls a step, in the order of a numeric progression.  The interval is B - A, but the “direction” is increasing from A to B. That this concept of number is distinct from the concept of fixed magnitudes of objects, where the number of fixed quantities in one collection may be equal to, greater than, or less than, another collection of objects, is of great significance to Hamilton.  He writes:

That a moment of time respecting which we inquire, as compared with a moment which we know, must either coincide with or precede or follow it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can comprehend an area. The notion or intuition of Order in Time is not less but more deep-seated in the human mind, than the notion of intuition of Order in Space; and a mathematical Science may be founded on the former, as pure and as demonstrative as the science founded on the latter. There is something mysterious and transcendent involved in the idea of Time; but there is also something definite and clear: and while Metaphysicians meditate on the one, Mathematicians may reason from the other.

Consequently, he proceeds to reason in this essay, which he calls a “Preliminary and Elementary Essay on Algebra as the Science of Pure Time.” In it, he develops the mathematical entities of a group representation under addition, though, of course, he doesn’t have any idea of that concept as it later develops. Yet, he introduces a new symbol to denote the inverse elements of the group, and one for the reciprocal operation (an upside down R, for Reciprocatio, which defines the operation for transforming an element of the group into its inverse.)  He also identifies the identity element of the group and the property of closure and associativity.

Moreover, he goes on to develop ratios of these numbers, as multiples and submultiples, defining the operations in a way very similar to the group under multiplication, which we have developed, although at this point I can only follow his development enough to see the general outlines of this.  Significantly, however, he refers to these numbers as “reciprocal numbers,” something which really blew me away, as anyone who has been following this blog would readily understand.

To say that the discovery of Hamilton’s “essay” on the mathematics of progression, as opposed to the mathematics of magnitude, is an important milestone in the work of the LRC, is an understatement of gigantic proportions. It is so startling that I’ve had to shift my priorities in order to devote time to studying it in detail, which is why the blogging has slowed down so much recently.

However, I promise to share more as soon as I can manage it.