Archives for category: Limits

Despite having already invented the Calculus (which he called the Theory of Fluxions), Newton did not use it in his magnum opus, the Principia Mathematica, probably because he felt uneasy about its logical basis. Instead he employed cumbersome strictly geometrical arguments without even employing co-ordinates ─ which makes the Principia almost unreadable for the modern student. Feynman, one of the greatest mathematical physicists of all time, confessed that he could not follow Newton’s proof that planets must follow elliptical orbits and instead offered his own geometrical proof (see Feynman’s Lost Lecture by Goodstein and Goodstein).
However, it is not true, as is often said, that Newton had no concept of limits. The very first section of Book I is entirely given over to eleven ‘Lemmas’ about Limits which he needs in order to show, amongst other things, that planets and other heavenly bodies verify an inverse square distance law. The key limit is the first:
“LEMMA I
Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.

 If you deny it, suppose them to be ultimately equal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is contrary to the supposition.”
Newton, Principia (Motte/Cajori translation  p. 29)

In particular, this Lemma leads on to the all-important Lemma VII which states that “the ultimate ratio of the arc, chord and tangent, any one to any other, is the ratio of equality”.

So, what are we to make of Lemma I? On the face of it, it sounds foolproof. Either diminishing ratios that converge to unity, attain their goal or they do not ─ exclusive sense of ‘or’. In practice, of course, this will not do; essentially Calculus wishes  to have it both ways, to make such ratios attain equality when this is convenient and have them not attain equality when this is embarrassing. At least Newton grasps the nettle: by this all-round Lemma he affirms that the limit is attained.
Or, does he? In the Scholium (Commentary) which concludes the section, Newton admits that there is a conceptual problem, at any rate when we consider speed. Why so? Because speed is not an independent entity but rather a ratio of distance to time, and, in dynamics, we desire to know a body’s speed at a particular moment of time. In such a case, is there, or is there not, such a thing as an ‘ultimate ratio’ of distance/time? Newton writes:

Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none”.

Newton’s reply to this objection is interesting:

By the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish…..There is a limit which the velocity at the end of the motion may attain, but not exceed.”

This is all very well but contradicts lemma I since Newton says in the above passage that this ‘ultimate ratio’ ‘may be attained’ ─ in which case it would constitute a difference D that is not supposed to exist according to lemma I.
And, a little further on, Newton even contradicts what he has just said since he now denies that this ‘ultimate ratio’ is in fact attained:
        Those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished ad infinitum” (p. 39 Motte/Cajori).

The contradiction remained like a worm in the apple of Calculus until the radical reworking the latter underwent at the end of the 19th century. The definition of a ‘limit’ that every mathematics student encounters today neatly sidesteps the problem ─ without resolving it. Mathematically speaking, it is immaterial whether a sequence or series actually attains the proposed limit; the only issue is whether the absolute value of the difference between all terms after a given point and the proposed limit can be made “smaller than any positive quantity” (Note 1).
The mathematics student of today is discouraged, sometimes even specifically prohibited, from asking the question that every enquiring person wants to pose: Does the function or sequence actually attain the limit? In most cases of any interest in Calculus and Analysis  the answer is that it does not. (The sequence 1, 1/2, 1/4, 1/8….1/2n for example does not ever attain the obvious limiting value of zero.) The adroit way in which the limit is defined, originally due to the 19th century mathematician Heine, means that, mathematically speaking, we get what we want, namely a clearcut test of whether or not a function ‘tends to a limit’ while avoiding altogether situations where, for example, we might find ourselves tempted to ‘divide by zero’.
However, Newton, despite being the greatest pure mathematician this country has  produced, was a physicist first and a mathematician second, which is why the modern ‘solution’ to the problem of limits, even had he thought of it, would probably not have appealed to him. I am afraid that I, as a philosophic empiricist, at any rate with regard to applied mathematics, am not at all satisfied by the sleight of hand; in cases of obvious physical importance I want to know whether a function or mode of behaviour generally actually does attain the proposed limit or not. However, Newton’s lemma VII which makes the “ultimate ratio of the arc, chord and tangent….the ratio of equality” does not convince me any more than it convinces any contemporary mathematics student.
So, what to do? The solution is quite simple and, I contend, perfectly valid mathematically ─ even though it will arouse howls of protest and derision from the aficionados of modern Analysis. We simply excise Newton’s lemma I and replace it by a positive statement:

“LEMMA I

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, do not in general become ultimately equal but differ from strict equality by a small but finite amount. 

        Now, it is true that in general we do not know what this ‘small amount’ is ─ although in most applications it either is or could conceivably be ascertained. We now know that all energy interactions are quantised and that the inevitable inefficiency (because of friction and similar considerations) of an actual machine can be (and often is) estimated. Not only that, calculus is already used in situations where we know the value of the independent variable cannot be arbitrarily diminished. For example, dn in molecular thermo-dynamics cannot be made smaller than the size of a single molecule and dx in population studies cannot be smaller than a single living person. This does not matter too much since we are dealing with millions of entities, although it must be said that in more accurate work, the tendency these days is to not bother with calculus but to slog it out numerically with computers to the degree of precision required.
This drastic pruning of calculus does not make Analysis, and all that depends on it, altogether redundant since there is often no great difference in practice between assuming that dx has an ‘ultimate’ final value and letting it go as near to zero as we wish ─ the dx terms and a fortiori second and third order terms will most likely end up by being discarded anyway. Nonetheless, one can and should question whether the assumptions of Analysis, especially infinite divisibility, are realistic. I believe they are not. There is a growing movement amongst physicists (e.g. Causal Set Theory, Loop Quantum Gravity &c.) that even spacetime, the last refuge of the devotees of the continuous, might be ‘grainy’.             SH  25/02/20

Note 1 The technical definition for a function is:
f(x) tends to a limit l as x tends to a, if, given any positive number ε (however small), there is a positive number δ (which depends on ε) such that, for all x, except possibly a itself, lying between a − δ and a + δ,  f(x) lies between l − ε  and l + ε .  The definition of the limit of a sequence is similar.
Such a definition will probably not mean much to the non-mathematical reader but the idea behind it is a sort of guessing game. I claim that some sequence or function tends to a limit l and my opponent challenges me to show that I can produce terms of my sequence or function that get me closer to this limit than some arbitrarily small quantity such as 10−6 = 0.000001. If I succeed, my opponent chooses an even smaller difference and so the contest goes on. The point is that this difference, though it can be reduced to zero in some cases, need not necessarily go to zero. For example, I might claim that the diminishing sequence 1; 0.1; 0.01; 0.001; 0.0001; and so on, has zero as a limit. My opponent asks me to get within 1/1000 of my limit, i.e. to make the difference d smaller than, say, 1/1000. I do this easily enough by presenting him with 0.00001 which is a term in the sequence but is smaller than  1/1000 (since 0.00001 = 1/10000). Moreover, since this is a strictly diminishing sequence, all  terms further down the line will also have a smaller difference than the one I have to better. If my opponent ups his challenge, I can easily meet it since if he comes up with 1/10N (for some positive integer N) I can get closer simply by adding more zeroes to the denominator. Yet, in such a case, if actually asked to produce a term in my sequence that makes the difference zero exactly I cannot do so ─ since any term 1/10N , however large N is, is still a positive quantity albeit a small one. But this does not matter, the limit still holds since I can get as close to it as I am required to.

 

 

 

 

Archimedes gave us the fundamental principles of Statics and Hydrostatics  but somehow managed to avoid founding the science of dynamics though, as a practising civil and military engineer, he must have had to deal with the mechanics of moving bodeis. The Greek world-view, that which has been passed down to us anyway, was essentially geometrical and the truths of geometry are, or were conceived to be, ‘timeless’ : they referred to an ideal world where spherical bodies really were perfectly spherical and straight lines perfectly straight.
Given the choice between exact positionaing and movement (which is change of position) you are bound to lose out on one or the other. But science and technology must somehow encompass  both exact position and ‘continuous’ movement. So how did Newton cope with the slippery idea of velocity ?  From a pragmatic point of view, supremely well since he put dynamics on a firm footing and went so far as to invent a new form of mathematics tailor-made to deal with the apparently erratic motions of heavenly bodies — his ‘Method of Fluxions’ which eventually became the Differential Calculus. Strangely, however, Newton completely avoided Calculus methods in his Principia and relied entirely on rational argument supplemented by cumbersome, essentially static,  geometrical demonstrations. Why did he do this? Probably, because he felt himself to be on uncertain ground mathematically and philosophically when dealing with velocity.
If you are confronted with steady straight line motion you don’t need Calculus — ordinary arithmetic such as even the ancient Babylonians and Egyptians employed is quite adequate. But, precisely, Newton was interested in the displacements of objects subject to a force, thus, by definition, not in constant straight line motion. And when the force was permanent, as was the case when dealing with gravitational attraction, the consequent motions of the boldies were never going to be constant (if change of direction was taken into account).
Mathematically speaking, speed is simply the first derivative of displacement with respect to time, and velocity, a vector quantity, is ‘directed speed’, speed with a direction attached to it. The modern mathematical concept of a ‘limit’ artfully avoids the question of whether a ‘moving’ particle actually attains a particular speed at a particular moment : it is sufficient that the difference between the ratio distance covered/time elapsed and  the proposed limit can be made “smaller than any finite quantity” as the time intervals are progressively reduced. This is a solution only to the extent that it removes the problem from the domain of reality where it originated. For the world of mathematics is an ideal, not real world though in some cases there is a certain overlap.
Newton was not basically a pure mathematician, he was a mathemartical realist and a hard-nosed materialist (at least in his physics). He was obviously bothered by the question that today you are not allowed to ask, namely “Did the particle attain this limit or did it only get very close to it?” 
It is often said that Newton did not have the modern mathematical concept of limit, but he came as close to it as was possible for a consistent realist. He speaks of “ultimate ratios” “evanescent quantities” and, unlike Leibnitz, tends to avoid infinitesimals if he can possibly manage to. He sees that there is indeed a serious logical problem about these diminishing ratios somehow carrying on ad infinitum and yet bringing the particle to a standstill.

“Perhaps it may be objected that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged that a body arriving at a certain place, is not its ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none.”
Newton, Principia    
Translation Andrew Motte

Note that Newton speaks of ‘at a certain place’ and ‘its place’, making it clear that he believes there really are specific positions that a moving particle occupies. He continues :

“But the answer is easy; for by ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place and with which the motion ceases. And in like manner,  by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish.”
  Newton, Principia     Translation Andrew Motte

      But this implies that there is a definite final velocity :  

        “There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity.”

         Well and good, but Newton now has to meet the objection that, if the ‘ultimate ratios’ are specific, so also, seemingly, are the ‘ultimate magnitudes’ (since a ratio is a comparison between two quantities). This would seem to imply that nothing can properly be compared with anything else or, as Newton puts it, that “all quantities consist of incommensurables, which is contrary to what Euclid has demonstrated”.  

     “But,” Newton continues, “this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of the ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.”   (Newton, Principia     Translation Andrew Motte)

       The last phrase (’till the quantities are diminished in infinitum’) seems to be tacked on. I was expecting as a grand climax, “to which they approach nearer than by any given difference, but never go beyond, nor in effect attain” full stop. This would make the ‘ultimate ratio’ something akin to an asymptote, a quantity or position at once unattained and unattainable. But this won’t do either because, after all, the particle does pass through such an ultimate value (‘limit’) since, were this not the case, it would not reach the place in question, ‘its place’. Bringing in infinity at the last moment (‘diminished in infinitum’ ) looks like a sign of desperation.
A little later, Newton is even more equivocal
“Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end.” 

But what meaning can one give to “quantities….diminished without end” ?  None to my mind, except that we need such quantities to do Calculus, but this does not make such concepts any more reasonable or well founded. The issue, as I said before, has ceased to worry mathematicians because they have lost interest in physical reality, but it obviously did worry Newton and still worries generations of schoolboys and schoolgirls until they are cowed into acquiescence by their elders and betters. The fact of the matter is that to get to ‘its place’ (Newton’s phrase) a particle must have a velocity that is the ‘ultimate ratio’ between two quantities (distance and time) which are being ‘endlessly diminished’ and yet remain non-zero.
In Ultimate Event Theory there is no problem since there is always an ultimate ratio between the number of grid positions displaced in a certain direction relative to the number of ksana required to get there. When doing mathematics,  we are not going to specify this ratio, supposing even we knew it : it is for the physicist and engineer to give a value to this ratio, if need be, to the level of precision needed in a particular case. But we know (or I do) that δf(t)/δt has a limiting value (Note 1)which we may call df(t)/dt if we so wish. Note, however, that the actual ‘ultimate ratio’ will almost always be more (or less) than the derivative since there will be non-zero terms that need to be taken into account. Also, the actual limiting value will vary according to the processes being studied since manifestly some event-chains are ‘faster’ than others (require less ksanas to reach a specified point).  Nonetheless, the normal derivative will usually be ‘good enough’ for practical purposes, which is why Calculus is employed when dealing with processes that we know to be strictly finite such as population growth or radio-active decay.   S.H.

Note 1 :      Why must there always be a limiting value?  Because δt can never be smaller than a single ksana — one of the basic assumptions of Ultimate Event Theory.

(The opening image is from a painting by Jane Maitland)         S.H.     5/08/12