Calculus and Imaginary Numbers

An engineering correspondent has written some very intelligent and informed objections to the essays "Philosophical Problems with Calculus" and "Imaginary Numbers." I think this is very revealing of the differences in method and purpose between mathematics and the sciences, on the one hand, and philosophy, on the other hand. I am bound to be a little defensive about accusations of not understanding mathematics, and in turn perhaps a little condescending about others who are unaware of philosophical issues; but I also think that one of the most interesting aspects of this exchange is that the correspondent is really not *free* of articulated philosophical views, which come out towards the end. It is hard to defend the idea that matters eliminated by the abstractions of mathematics and science can be "laid to rest" and forgotten without making some epistemological claim that is *not* part of mathematics or science themselves, but actually part of a philosophical meta-theory about the natures of these disciplines. Practicing mathematicians and scientists can certainly be excused for regarding "metaphysics" as just meaning "nonsense," and for taking naturally to philosophical principles (like Logical Positivism) that affirm this. All e-mails are given in their entirety.

This page used to give the name and e-mail address of the correspondent. These have now (2003) been removed to protect his privacy.

[17 August 1999]

Dear Professor Ross,

While searching for algorithms for comparing various calendar schemes I came across your web pages. I read them with great enthusiam, particularly the sections on philosophy, economy, society and religion. It is refreshing to read philosophical topics in plain language and with an openess and fairness one quite often misses with most authors.

As a scientist, however, I have problems with "problems" from my field of knowledge. Take for instance your "Philosophical Problems with Calculus": The way derivatives are explained in this essay is simply not the way they are presented in modern teaching.

Let us start at line VI: On the left side of the equation sign is a ratio of finite quantities, nothing strange there, all well behaved. This ratio is a function of two variables, i.e. of x and of delta_x as evidenced by the right hand side of the equation. Let us call this function z. It can be plotted in a perspective graph as height over a plane; it will look something like mountains and valleys. Now one can trace various paths in this landscape. Among the infinity of selectible paths will be one along which the variable delta_x is constantly zero. This particular path is called the derivative. So where are the philosophical problems? There is no division by zero anywhere on the right hand side of equation VI which defines the function z! The notation delta_y over delta_x instead of z is short hand notation with purely historical reasons (Leibniz) (not unlike the strange ways English words are spelled) it has nothing to do with division by zero. As you surely know, there are many other styles for denoting derivatives such as primes and dots (Newton), subscripts (analytic geometry), operators (nabla of vector and tensor analysis), capital D in differential equations, and others.

Your attempt to represent infinitesimals by infinitely long decimal fractions is mistaken, because infinitesimals are an entire new class of mathematical objects. Trying to catch them by approximating them with real numbers leads to the "results" of the last paragraph of the essay.

Zero divided by zero has in a an overwhelming number of practical cases a well-defined value: An example to be found in any textbook on calculus is sin(2x)/x which becomes 2 as x approaches 0. But you know that, I am sure.

Trying to follow your ideas on imaginary numbers I got equally lost. What is the meaning of "exist" in this context? Whatever meaning the word exist is endowed with, it applies equally to all objects of mathematical investigations (not yet, of course, of unsettled mathematical investigations). Natural numbers (i.e. positive integers) do not exist in any higher or better way than imaginary numbers or quaternions or third order tensors. Like those entities they are generated by axioms, in the case of positive integers by the axioms of Peano. One may have the illusion that they are more "real" than imaginary numbers because one can map the set of natural numbers on a herd of cows or pills in a box. And cows and pills are very tangible and every-day-ish. But negative integers can be mapped on debts, complex numbers which are made up of real and imaginary numbers can be mapped on rotations of geometric objects in the plane, quaternions can be mapped on rotations in space, tensors can be mapped on elastic deformations of solids and so on. (Incidentally, it took nearly two hundred years to get negative numbers universally accepted because of the difficulties of mapping them on cows in the pasture). Trying to derive imaginary numbers from real numbers by a process of complete induction must fail because the imaginary element is an addition to the field (technical term) of real numbers, not contained in the field of real numbers and hence cannot be based on it. A fortiori, this failure cannot be used to demonstrate that imaginary numbers "exist" in a lesser sense than real numbers. -- All this can be found in an introductory level book on the theory of numbers; a famous title is *Algebra* by L. Van Der Waerden.

Many misunderstandings (I think I talk here about misunderstandings; most, if not all, human problems come from misunderstandings, not the least linguistically ones) come from technical terms which arose for historical reasons and caused unforseen difficulties with the public: a good example is Theory of Relativity; this term caused considerable hostilities towards the sciences, particularly here in Germany, because "relativity" was equated with arbitrariness and loose morals. Nobody would have cared if the theory would have been called Gauge Transformation of Electromagnetic Fields. As a student I witnessed a bodily attack on Heisenberg by a protestant priest who accused Heisenberg at the top of his voice of corrupting youth. I guess that terms like "real", "imaginary", "infinite", "rational", "irrational", "transcendental", "defective", "rank" may convey emotional connotations to those who are not active in the respective fields of knowledge.

Finally, a remark on something quite external to my field, but puzzling to me: Kant and Schopenhauer seem to get quite a lot of attention these days. For instance, Brian Magee spends a quarter of his ca. 500 pages of *Confessions* on these two philosophers. Your whole edifice of philosophy seems to rest to a substantial extent on their ideas. How come that recent developments in the life sciences touching upon epistemology do not even get mentioned, let alone dealt with? I am referring to Konrad Lorenz, Gerhard Vollmer, Rupert Riedl, Valentin Braitenberg and many others who have shown that *a priori* knowledge is a posteriori use of phylogenetically acquired facilities. All these authors explicitely refer to Kant and many to Schopenhauer, sometimes over many pages, and demonstrate a remarkable knowledge of their ideas. Are the philosophers not afraid that solutions to some of their problems are already laid down in the publications of life sciences? (After having read Magee I personally am convinced that half of Kant's open questions have been adequately answered by the sciences, particularly by neurophysiology).

Sincerely

[17 August 1999]

Dear Mr. XXXX,

Thank you very much for your feedback and lengthy reponse to some of my pages on mathematics. I shall try to respond to each comment.

At 01:28 PM 8/17/99 +0200, you wrote:

As a scientist, however, I have problems with "problems" from my field of knowledge. Take for instance your "Philosophical Problems with Calculus": The way derivatives are explained in this essay is simply not the way they are presented in modern teaching.

First, remember that the "Problems with Calculus" is first of all about *historical* objections to calculus, starting in Newton's day. However it may be that calculus is presented in modern teaching, that is not relevant to the point. Second, although objections to infinitesimals were diverted for a long time with the theory of limits, I was intrigued that a theory of infinitesimals reemerged in non-standard analysis, by the addition of extra axioms to standard set theory. Thus, if infinitesimals actually were necessary to calculus, and if the objections were cogent, this nevertheless can all at once be overcome. An important point for the philosophy of science or of mathematics is that this really didn't make any difference anyway. Even if the objections to calculus were cogent, somehow calculus worked anyway, and the objections did not affect that. Third, I presented derivatives in the way I did because that treatment, from an older textbook, made more sense to me that more recent treatments I have seen. And since the older treatment highlights the issue of infinitesimals, that is helpful in a discussion of the objections to infinitesimals in Newton's day.

It can be plotted in a perspective graph as height over a plane; it will look something like mountains and valleys. Now one can trace various paths in this landscape.

This is a model. Whatever a model or a map shows, this avoids the philosophical or metaphysical questions about the thing itself.

Among the infinity of selectible paths will be one along which the variable delta_x is constantly zero. This particular path is called the derivative. So where are the philosophical problems?

The philosophical question is about the thing **from which** the map is modeled. It is the delta_x being zero that is the problem in the **original** context. Your impatience with this may be due to the fact that is not relevant for science, or even, to a great extent, for mathematics. A mathematical argument can stipulate or postulate anything, without examining, or even much caring, what it really means to do so. The logical consequences are all that count. In science, if you can make a model or a map or a representation **in any way** and that gives good results, then there is no problem. On the other hand, other people would like to know about the thing itself, not about something that works, in some respect, like it. Also, the metaphysical issues that get swept aside can also be clues about the future of science itself. infinitesimals were avoided in response to objections to them, but later a way turns up of representing them. Objections to Newtonian action-at-a-distance were quietly forgotten as the theory otherwise was so successful, but neither quantum mechanical nor Relativistic modern treatments of gravity are theories of action-at-a-distance, deftly reviving the issue and solving it at the same thing -- though with a minor inconvenience that gravity is mediated by different things in each theory.

The fundamental question is about abstraction. Abstraction leaves thihgs behind. Mathematics itself is a vast abstraction, leaving behind colors, cold beer, hot dates, etc. In every case, however, what is left behind does not disappear. One can hope that what is left behind can eventually be treated by means of the abstractions, as the contour of mountain ranges could surprisingly be treated with fractals, but this is really no more that a hope or a postulate. Kant could give it more dignity by saying that it is a rational postulate of science (the "regulative employment" of Reason), but scientists, for the practice of science, don't have to care what it is, though occasonally some of them get curious.

There is no division by zero anywhere on the right hand side of equation VI which defines the function z!

True, but that is not the point. The right hand side of the equation is made equivalent to the left hand side, which does have the division by zero (if delta_y and delta_x are reduced to zero). This gives a finite value for 0/0, which is why the technique works, but it doesn't address the paradox of the left left.

The notation delta_y over delta_x instead of z is short hand notation with purely historical reasons (Leibniz) (not unlike the strange ways English words are spelled) it has nothing to do with division by zero.

First, the notation is incidental to the operation that is involved. Second, your argument proves too much: In Newton's day, infinitesimals were proposed to **avoid** division by zero. If you are right, then nobody understood that there was no division by zero involved. (The "historical reason" is not just an artifact of notation, but a characteristic of how the matter was understood.) Instead, the theory of limits, which was accepted as allowing the avoidance of infinitesimals, was acceptable because it too was able to avoid the real reduction of delta_x to zero and so division by zero. Delta_x does not reach zero but "approaches it as a limit." Your argument makes this whole history senseless, since I understand you to say that there was no problem in the first place. I can hardly discuss the history without taking into account the objections and responses at the time, and the real reasons for them.

Your attempt to represent infinitesimals by infinitely long decimal fractions is mistaken, because infinitesimals are an entire new class of mathematical objects.

They are now that there is an axiomatic treatment of them. There was **not** in Newton's day. infinitesimals were essentially a metaphysical theory, and this circumstance is well revealed by the fact that they were forgotten for a long time without loss. My question was simply how, naively, an infinitesimally small number could be represented as a decimal. This has the interesting result that it seems to be, by ordinary algebraic methods, equal to zero. This has nothing to do with infinitesimals as they can now be treated in set theory. They are still smaller than any number that can be written, and so equal to zero in any practical sense.

Trying to catch them by approximating them with real numbers leads to the "results" of the last paragraph of the essay.

That was the point.

Zero divided by zero has in a an overwhelming number of practical cases a well-defined value: An example to be found in any textbook on calculus is sin(2x)/x which becomes 2 as x approaches 0. But you know that, I am sure.

Note: I have no doubt that zero divided by zero has well defined values and practical uses. But if it does, then your objection that derivates are **not** zero divided by zero is kind of unnecessary. Derivates would not be threatened, as in fact they were not. What remains of interest about 0/0 is that it can have **any** value, depending on how it is derived.

Trying to follow your ideas on imaginary numbers I got equally lost. What is the meaning of "exist" in this context? Whatever meaning the word exist is endowed with, it applies equally to all objects of mathematical investigations (not yet, of course, of unsettled mathematical investigations). Natural numbers (i.e. positive integers) do not exist in any higher or better way than imaginary numbers or quaternions or third order tensors.

Saying that "all objects of mathematical investigation exist" may be a convenient postulate for mathematicians; but, in any referential or metaphysical sense, is it not self-evident and is not even an argument. The specific problem with the "existence" of imaginary numbers (meaning the real objects they would refer to) is that they are defined by way of a contradiction: those real numbers whose square roots are negative. Since there are no real numbers whose square roots are negatives, the name "imaginary" was proposed and used for a quite understandable reason. With the postulate that "all objects of mathematical investigation exist," then it will be trivially true that imaginary numbers exist, but this simply begs the question.

Like those entities they are generated by axioms, in the case of positive integers by the axioms of Peano.

Axioms are not the ontological ground of numbers, only their logical ground. What numbers are metaphysically is entirely untounched by axiomatization, just as the nature of space itself was untouched by Euclid. Since both numbers and space were known before Euclid, let alone before Peano, they were certainly not even originally **known** through axioms. So the axioms are not epistemologically fundamental either.

One may have the illusion that they are more "real" than imaginary numbers because one can map the set of natural numbers on a herd of cows or pills in a box. And cows and pills are very tangible and every-day-ish.

They are, are Aristotle put it, "primary substances." "Every-day-ish" things have the priority of being concretely existing things, through which we live and die, not abstract objects which causally affect nothing. Non-concrete objects raise metaphysical questions in general. No less so the abstract objects of mathematics.

But negative integers can be mapped on debts, complex numbers which are made up of real and imaginary numbers can be mapped on rotations of geometric objects in the plane, quaternions can be mapped on rotations in space, tensors can be mapped on elastic deformations of solids and so on.

Again, as I said above, "mapping" transfers and avoids a question but does not anwser it. Abstracting from a question may enable a theory to work, about something else, but it does not make or prove that the unabstacted features are meaningless or unimportant. The entire essay on imaginary numbers (http://www.friesian.com/imagine.htm) was addressing Isaac Asimov's argument that mapping imaginary numbers all by itself makes them as real as anything else. The only way to ultimately maintain that position is to argue that nothing means anything and that structures of abstract representation do not need to be interpreted (given a meaning) because nothing does. This a serious and sustainable position to take, which is why Roger Penrose's argument that Gödel's Proof contradicts formalism is of great importance.

Trying to derive imaginary numbers from real numbers by a process of complete induction must fail because the imaginary element is an addition to the field (technical term) of real numbers, not contained in the field of real numbers and hence cannot be based on it. A fortiori, this failure cannot be used to demonstrate that imaginary numbers "exist" in a lesser sense than real numbers. -- All this can be found in an introductory level book on the theory of numbers; a famous title is Algebra by L. Van Der Waerden.

It is also a question-begging argument. If the real numbers are alone real, then anything not contained in the field of real numbers is going to be, by definition, **non-real**. If this is to be disputed, then some argument must establish that something besides the real numbers is, in fact, real. The only way you do this is with the postulate, evidently, that anything derived from number axioms (or the practice of mathematics) is real, which is not even relevant to the question (you would have to be St. Anselm to believe that existence is established by a priori axioms). Since you allow yourself that real numbers, even negative numbers can be applied ("mapped") onto cows and such (though they are not "mapped," but simply *refer*), you should appreciate that, if cows and such are *prima facie* real, then real numbers are what are about real things. "Mapping" imaginary numbers is essential for you, because that is the only way to abstract from and avoid the contradiction by which imaginary numbers are defined. This gets you nowhere even near the argument of the "Imaginary Numbers" essay, which is that, because of Kantian ontology, imaginary numbers can exist, in a significant sense, even if their objects are self-contradictory.

Many misunderstandings (I think I talk here about misunderstandings; most, if not all, human problems come from misunderstandings, not the least linguistically ones) come from technical terms which arose for historical reasons and caused unforseen difficulties with the public:

Here the misunderstanding is about the difference between abstraction (and methods of science and mathematics) and about the questions of existence and meaning that can easily be deferred by science but that remain of interest to people in general, to philosophers, and even to scientists who are concerned with what the ultimate questions and answers are going to be like ("God is subtle but not malicious"). Your view that the only misunderstandings are over historical accidents of terminology ignores the metaphysics entirely and also ignores the history where the issues were not terminological, but philosophically (if not scientifically) substantive.

a good example is Theory of Relativity; this term caused considerable hostilities towards the sciences, particularly here in Germany, because "relativity" was equated with arbitrariness and loose morals.

Since many people were already convinced of moral relativism, they liked the idea that something in science supported their viewpoint. Some hostility to Einstein followed from the belief that the science **did** support the moral relativism. That the most important feature, indeed the logical foundation, of Relativity, was an Absolute, the velocity of light, that the whole business was irrelevant to morality anyway, and that Einstein himself was no moral relativist, were are all features that could be overlooked.

Nobody would have cared if the theory would have been called Gauge Transformation of Electromagnetic Fields. As a student I witnessed a bodily attack on Heisenberg by a protestant priest who accused Heisenberg at the top of his voice of corrupting youth.

This may have been more because of Heisenberg's association with the Nazis and with "Aryan science" during the Hitler era.

I guess that terms like "real", "imaginary", "infinite", "rational", "irrational", "transcendental", "defective", "rank" may convey emotional connotations to those who are not active in the respective fields of knowledge.

If they are to have more than a formalistic, functional, or pragmatic "connotation," then the relevant field of knowledge is actually metaphysics. Saying "emotional connotations" simply means that you don't know why anyone ever worried about it.

You do not, indeed, have to worry about it, since none of it need be relevant to your work. Since I am not a mathematican or a scientist, the essays I write are about the metaphysical and other philosophical issues. If these simply don't mean anything to you, then you can happily ignore them, or think that I simply don't understand the math. I really think that some kind of questions simply don't register for some people, and contempt for philosophy, from people who don't see the point, is nothing new or unusual.

Finally, a remark on something quite external to my field, but puzzling to me: Kant and Schopenhauer seem to get quite a lot of attention these days.

That would be nice, though it certainly is not enough attention.

For instance, Brian Magee spends a quarter of his ca. 500 pages ofConfessionson these two philosophers. Your whole edifice of philosophy seems to rest to a substantial extent on their ideas. How come that recent developments in the life sciences touching upon epistemology do not even get mentioned, let alone dealt with?

Because they are not relevant. Philosophical questions about knowledge are very different from cognitive science questions about knowledge. Indeed, John Searle, someone much more familiar with cognitive science than I am, in his recent *The Rediscovery of the Mind* [MIT, 1992], actually argues that the whole project of cognitive science is misconceived. I don't know whether it is or not, but the circumstances of an empirical and experimental approach to knowledge mean that the philosophical problem of knowledge is bypassed.

I am referring to Konrad Lorenz, Gerhard Vollmer, Rupert Riedl, Valentin Braitenberg and many others who have shown thata prioriknowledge is a posteriori use of phylogenetically acquired facilities.

"Shown" is always an interesting word when used among all the uncertainties of scientific results, which is why scientific results are always a mixed blessing in philosophical arguments.

All these authors explicitely refer to Kant and many to Schopenhauer, sometimes over many pages, and demonstrate a remarkable knowledge of their ideas.

Anyone, like Chomsky, who thinks that there is an innate component to human knowledge or cognitive functions also can appeal to the philosophical precedents, like the Rationalists, or Kant. This is always a mixed blessing for scientists, whose results cannot be *justified* by the philosophical precedents and who are liable to become tangled in philosophical issues that they can't answer and don't even want to deal with. It is all rather like being on a train, looking out the window, and seeing the most beautiful and appealing woman imaginable through the window on another train passing in the opposite direction (not high speed trains, of course!). However perfect and desirable, there is almost no possible way to get to her, or to find her again in the future. The appropriate thing to do is have a look at the women on one's own train.

Are the philosophers not afraid that solutions to some of their problems are already laid down in the publications of life sciences? (After having read Magee I personally am convinced that half of Kant's open questions have been adequately answered by the sciences, particularly by neurophysiology).

Kant's basic question is about synthetic propositions *a priori*. Even if we in fact believe such propositions because of physiology, which is what Hume already thought, this is not relevant to the question of necessary truth, as Hume also understood. Should physiology affirm Hume in the former respect, the latter issue is still just as open as ever.

Best wishes,

Kelley L. Ross

[18 August 1999]

Hello, Professor Ross,

Thanks for your fast and exhaustive answer. It became very clear to me (again) that the gap between the sciences and the humanities is really very vast. In your mail you point out that some (most, all) points raised are historically motivated. This was not clear to me from reading the original essays. So I will not comment any further on our disagreements, I will only try to clarify some of the stuff I wrote.

The philosophical question is about the thingform whichthe map is modeled. It is the delta_x being zero that is the problem in theoriginalcontext.

My point is that the problem was legitimately transformed into a perfectly well-behaved equation:

My remarks relate to this equation which is valid independently of how it was deriveds.There is no division in sight, hence there is no problem with division. I maintain that z is another name for a rational, finite number which was constructed by dividing a finite delta_y by a finite delta_x, no divisions by zero, no infinitesimals, no paradox. My image of a landscape was meant to emphasize the "normality" of the function z. I fail to realize that there is a metaphysical (in fact, I do not know what "metaphysical" means) problem, swept aside or not. In recounting history you are absolutely right, but I believe that at least some historically grounded philosophical (in those days including the natural sciences) questions have been adequately answered and may be laid to rest.

The specific problem with the "existence" of imaginary numbers (meaning the real objects they would refer to) is that they are defined by way of a contradiction: those real numbers whose square roots are negative.

The point is, they are not **real** in the mathematical definition of the word. In algebra on the level we are talking, all objects are defined as solutions of equations:

positive integers -> defined by axiomsnegative numbers solve a + x = b where b < a; a, b positive integers

rational numbers solve a x = b where a, b are integers

real numbers solve x x = a where a >= 0 and rational

imaginary numbers solve x x = a where a is anything; (what is contractictory about that?).

There are no positve numbers whose sum is zero. There are no natural numbers > 1 whose product is 1. There are no rational numbers whose square is 2, and so on. In other words, what you say about imaginary numbers is similarly true for all other kinds of numbers save the positive integers.

What numbers are metaphysically is entirely untounched by axiomatization, ...

There is the core of the mutual misunderstandings: I don't know what metaphysical questions are.

... Isaac Asimov's argument that mapping imaginary numbers all by itself makes them as real as anything else. The only way to ultimately maintain that position is to argue that nothing means anything and that structures of abstract representation do not need to be interpreted (given a meaning) because nothing does.

This sequitur leaves me baffled.

It is also a question-begging argument. If the real numbers are alone real, then anything not contained in the field of real numbers is going to be, by definition,non-real.

This argument and what follows is a confusion of the mathematical term **real** and the philosophically used term **real**.

Your view that the only misunderstandings are over historical accidents...

I said "many".

This may have been more because of Heisenberg's association with the Nazis and with "Aryan science" during the Hitler era.

No, the priest was Dr. Sapper of Graz, Austria, who became notorious for his pamphlets against relativity. He was not known to have come out against Naziism.

Saying "emotional connotations" simply means that you don't know why anyone ever worried about it.

No, it simply means that I do not know why anyone still worries...

Because they are not relevant.

This statement would register better if philosophers discussed the (non-)relevancy.

"Shown" is always an interesting word when used among all the uncertainties of scientific results, which is why scientific results are always a mixed blessing in philosophical arguments.

This is certainly true. But the history of human knowledge shows (again that four-letter word) that some areas of philosophical investigations migrated to sciences. This process seems to me to be a one-way street.

of the question of necessary truth,

What is truth? Only an observer outside of both the human mind and the external world could make that decision. Lacking such an observer, we should entertain the heuristic assumption that one philosophy (yes, philosophy) is more "true" than another if it is to a higher degree coherent and more successful with predictions than any rival theory. Every other stand, particulary one which relies on introspection, is extremely suspect.

By the way, I share your enthusiasm for calendar systems. I am just finishing a software calendar for the palmtop computer PSION. I am looking for a comprehensive description of the calendar of the French Revolution; do you know one?

Kind regards,

[18 August 1999]

Dear Mr. XXXX,

I might mention that your e-mail seems to get reformated at least a couple times and arrives in a very chopped up form, making it very hard to read. You might CC your e-mail to another of your own e-mail addresses to see how it looks.

At 10:38 AM 8/18/99 +0200, you wrote:

It became very clear to me (again) that the gap between the sciences and the humanities is really very vast.

The gap is not between science and the humanities but between scientific method and the general logical and epistemological criteria of philosophy. You will notice that I have not said that what you do is wrong or meaningless, or that calculus or imaginary numbers are somehow illegitimate or should not be used. But you have accused me of misunderstanding mathematics and of trespassing into your speciality. Not so. I deal with philosophical issues that attend mathematics, meta-mathematical issues. By attacking or dismissing those, you trespass into my speciality -- though this is no sin, just a misunderstanding. Indeed, I think it finally comes out in this e-mail that you are not innocent of philosophical views, but you have philosophical preconceptions or preferences that motivate your dismissals. It is your philosophical position, not your mathematical practice, that is at odds with my essays.

In your mail you point out that some (most, all) points raised are historically motivated. This was not clear to me from reading the original essays.

The essays were about the issues of historical importance. Your response seems to be "Oh, we can forget about those now." You can work that way in science, because some things just die and get buried. But the philosophical questions don't work quite the same way and are not so easily buried.

My point is that the problem was legitimately transformed into a perfectly well-behaved equation:z(x, delta_x) = 6x + 3 delta_x + 4

This is the method of abstraction that I mentioned. If you toss out the original ("historical") consideration of y and delta_y and the ratios y/x and delta_y/delta_x, then you have abstracted from the attendant issues. In the equation you have written, the left side is simply another, more abstract, way of writing the right side, making the equation an empty tautology (as though to say, "'6x + 3 delta_x + 4' is a function of x and delta_x"). This completely loses the original left side and erases the logical means and considerations (not just the trivial accidents of history) by which the right side of the equation was derived.

My remarks relate to this equation which is valid independently of how it was deriveds.

Indeed. By the same token z(x, delta_x) = w(x, delta_x) is also "valid independently of how it was derived," but it tells us even less. It is still **true**; there is just a significant loss of meaning. That is not just historical loss, but semantic loss.

There is no division in sight, hence there is no problem with division.

As I said previously, abstracting from something doesn't mean it is meaningless or can be forgotten forever. As practical matter in science, it often **can** be forgotten, because of the nature of the method.

I maintain that z is another name for a rational, finite number which was constructed by dividing a finite delta_y by a finite delta_x, no divisions by zero, no infinitesimals, no paradox.

There can still be infinitesimals, because the delta_x is still a factor. If dy/dx = 6x + 4, then how delta_x gets to 0 is still of interest.

Actually, I'm not sure why you are arguing with **me** about this. Infinitesmals are no longer a historical curriosity but, as I've mentioned more than once, and you have acknoweldged, have been revived. That was done for a reason, and the discussions I have seen of it explain it in the traditional context of conceptual problems over analysis.

In recounting history you are absolutely right, but I believe that at least some historically grounded philosophical (in those days including the natural sciences) questions have been adequately answered and may be laid to rest.

Again, laying then to rest in science is not the same thing as settling the problem philosophically. Nor is it entirely irrelevant to science, since some old issues have a way of turning up again, as I have mentioned.

imaginary numbersThe point is, they are not [?]

realin the mathematical definition of the word.

As I said, the mathematical definition(s) you mentioned beg the metaphysical question.

In algebra on the level we are talking, all objects are defined as solutions of equations:

To talk about reality (e.g. the reality of imaginary numbers), this requires the metaphysical postulate that "All solutions of equations have a referent in existence." Why this would be so raises all the interesting metaphysical, meta-mathematical questions. So say that "all solutions of equations" are "real" by definition, in any substantive sense begs the question.

positive integers -> defined by axioms

Again, "defined by axioms" tells us nothing either epistemologically or ontologically. It simply says, "The axioms of set theory are logically sufficient to the positive integers." Wow. Then, as Aristotle did, we ask, "What makes the axioms true, or their referents 'real'?" The other definitions you give (negative numbers, etc.) presuppose the "solution of equations" axiom above.

imaginary numbers solve x x = a where a is anything; (what is contractictory about that?).

Since imaginary numbers are solutions to equations, but we know nothing about the reality of the referents of solutions of equations, the metaphysician then asks "But what **are** the imaginary numbers?" Which leads back to the traditional discussion of their definition in terms that are not so abstracted.

There are no positve numbers whose sum is zero. There are no natural numbers > 1 whose product is 1. There are no rational numbers whose square is 2, and so on. In other words, what you say about imaginary numbers is similarly true for all other kinds of numbers save the positive integers.

You simply list things that don't exist. Square circles don't exist. I never said that the problem of imaginary numbers simply means that they can't be used or must be forgotten about. Since they are solutions to equations and can be the means to real solutions for physical events, they clearly have more existence than "positive numbers whose sum is zero," though the paradox that attends them is, indeed, that "the square of no positive or negative number, or zero, is negative." The essay on imaginary numbers was to provide a metaphysics for imaginary numbers. You simply don't think that is necessary. But that is clearly because you think that saying "solution to an equation" is sufficient metaphysics. But it would be **only** with a number of unexamined, even unadmitted postulates.

What numbers are metaphysically is entirely untounched by axiomatization, ...There is the core of the mutual misunderstandings: I don't know what metaphysical questions are.

Indeed. In the simplest terms, metaphysics (or ontology) is about the nature of existence. "Mathematics exists" is a true statement, but it is an empirical statement and not part of metaphysics. "The objects of mathematical representation exist" is a statement both of meta-mathematics and metaphysics. That metaphysical statement can be avoided with principles such as "representations do not refer," i.e. no knowledge **has** an object. This would mean that there is no such thing as "objects of mathematics" and we can simply say that "imaginary numbers exist" because "imaginary numbers are part of mathematics" and "mathematics exists" -- which I think is your move.

... Isaac Asimov's argument that mapping imaginary numbers all by itself makes them as real as anything else. The only way to ultimately maintain that position is to argue that nothing means anything and that structures of abstract representation do not need to be interpreted (given a meaning) because nothing does.This sequitur leaves me baffled.

It is fragment of and reference to the kind of arguments made by Wittgensteinians and deconstructionists against representation, truth, and non-natural meaning. So it is about questions that don't seem to interest you or that you don't seem to think are even necessary.

It is also a question-begging argument. If the real numbers are alone real, then anything not contained in the field of real numbers is going to be, by definition,non-real.This argument and what follows is a confusion of the mathematical term

realand the philosophically used termreal.

I beg your paradon: Not a "confusion," a distinction. You are the one confusing them. The "mathematical term **real**," as you have defined it, is simply a stipulation -- whatever you do as a mathematician is "real." The "philosophically used term **real**" is about what exists or what exists independently of us, which means we must ask, not stipulate, "What is **real**?" The problem is that you may regard the stipulation (or postulate) of your method as actually answering the philosophical question. It doesn't.

This may have been more because of Heisenberg's association with the Nazis and with "Aryan science" during the Hitler era.No, the priest was Dr. Sapper of Graz, Austria, who became notorious for his pamphlets against relativity. He was not known to have come out against Naziism.

Then this Dr. Graz seems to be a confused person.

Saying "emotional connotations" simply means that you don't know why anyone ever worried about it.No, it simply means that I do not know why anyone still worries...

People worry about the metaphysical questions that you do not, evidently, worry about. That's OK, but you don't have any business complaining to me about it.

Because they are not relevant.This statement would register better if philosophers discussed the (non-)relevancy.

Logically, "relevant" means "germane to the question or the argument." The issue here is the difference between metaphysical questions and the practice of mathematics. Your view seems to be that the practice of mathematics settles the metaphysical questions, which it logically does not.

"Shown" is always an interesting word when used among all the uncertainties of scientific results, which is why scientific results are always a mixed blessing in philosophical arguments.This is certainly true. But the history of human knowledge shows (again that four-letter word) that some areas of philosophical investigations migrated to sciences. This process seems to me to be a one-way street.

"Migrated" is not an all-or-nothing proposition, so it is not entirely a one-way street. When scientists or mathematicians say that results are "elegant" or "beautiful," they are no longer speaking as scientists, and not even as philosophers, but as artists. The philosophical question is then what "elegant" or "beautiful" have to do with science and nature and why scientific results are spoken of in that way. More importantly, in "migrating" to science, a discipline (like cosmology) can carry some philosophical issues along with it. Even a complete migration, however, opens the door to the meta-questions. Meta-mathematics, in dealing with the foundations and metaphysics of mathematics, is largely part of philosophy.

of the question of necessary truth,What is truth?

Asked Pilate.

Only an observer outside of both the human mind and the external world could make that decision.

False.

You sound like a hopeless Cartesian, epistemically locked in your head.

Lacking such an observer, we should entertain the heuristic assumption that one philosophy (yes, philosophy) is more "true" than another if it is to a higher degree coherent

Only for a "coherence" theory of truth.

and more successful with predictions than any rival theory.

Only with the logical use of falsification, as in scientific method.

Every other stand, particulary one which relies on introspection, is extremely suspect.

"Every other stands" implies that you know what "every other stand" is, which you fairly clearly do not.

Here is some clue about the background of philosophical presuppositions that is the true source of your complaints. I am not interested in critiquing your philosophical statements, except to the extent that I have indicated, since I have not written to complain about you, only to explain what you have been complaining about. I don't have the time or desire to do otherwise. If you are really interested in philosophy, you can pursue you own investigations, examine the relevant material at the *Proceedings of the Friesian School* site, and write to me separately about it.

Yours truly,

Kelley L. Ross

[19 August 1999]

Hello again, my answer to your most recent reply is in the attachment, and therefore, hopefully, more readable.

Sincerely,

[attachment:]

Dear Prof. Ross,

Your remarks in your first three paragraphs of your most recent letter are very kind and understanding; I appreciate that, sincerely, although I fail to go along with you with many other topics of our conversation. As an industrial researcher with an extensive training in mathematics I thought you made statements about mathematics which I simple could not recognize as what they were supposed to be, i.e. metaphysical statements.

The essays were about the issues of historical importance. You response seems to be "Oh, we can forget about those now." You can work that way in science, because some things just die and get buried. But the philosophical questions don't work quite the same way and are not so easily buried.I always thought that all intellectual activities are directed to reaching answers to questions; apparently (judging from reading a number of authors recently) in philosophy the question is more important than the answer.

There can still be infinitesimals, because the delta_x is still a factor. If dy/dx = 6x + 4, then how delta_x gets to 0 is still of interest.In my equation z( ) = ... delta_x is a number which may be set to any value, including zero; it is not an infinitesimal!

"Every other stand" implies that you know what "every other stand" is, which you fairly clearly do not.If I say: "This is my mother, no other woman is my mother", do I have to know every woman in the world to render this statement true? If I say "There is no perpetual motion machine", do I have to examine every proposal for such a machine to render the statement true?

Again, thank you for the interesting conversation. Like you, I do not wish to continue it because the common ground is too narrow. Maybe I am a hopeless case.

I wish you well for the future and lots of success!

Sincerely

[19 August 1999]

Dear Mr. XXXX,

Very briefly, to answer you last comments.

I always thought that all intellectual activities are directed to reaching answers to questions;

Truly.

apparently (judging from reading a number of authors recently) in philosophy the question is more important than the answer.

No, but many philosophical questions, especially in ethics and metaphysics, are much more difficult to answer, especially when first the arguments of many people that knowledge in ethics and metaphysics (or even anywhere) is impossible, or even meaningless, must be disposed of. That is why mathematics and science can be so powerful in their own areas, since so much messy detail can be left behind. It is my business to keep looking at the messy detail, when it is relevant to other continuing questions, e.g. metaphysical ones.

If I say: "This is my mother, no other woman is my mother", do I have to know every woman in the world to render this statement true?

No, but it is still important who you mother is and what she is like, which we do not know from these statements. This illustrates my point rather well. The statements say nothing about your mother. When people ask about your mother, do you really just say "This (or that) is my mother," just because more informative statements can be reduced to something like that? Of course not, even though what you say is certainly TRUE. "This" is a demonstrative, and if you were pointing at your mother, at least I could see her, and see what she looks like; but we are removed even from that situation, so the demonstrative actually adds nothing to the bare expression "my mother."

Also, in ordinary situations (apart from adoption, host mothering, etc.), it does not add any additional information to say "no other woman is my mother," since we usually presuppose that people only have one (birth) mother. So we are at a level of abstraction that is so high, that the semantic content all but vanishes. As you say, other women in the world are not very relevant to knowing your mother, but something substantive about your mother is relevant to knowing about your mother.

Again, thank you for the interesting conversation. Like you, I do not wish to continue it because the common ground is too narrow. Maybe I am a hopeless case.

The conversation is interesting enough that I plan to post it with the Correspondence of the Proceedings (at http://www.friesian.com/corespnd.htm). If you do not want your name to appear, let me know. If you have a webpage that you would like linked, let me know and that will be included.

Best wishes,

Kelley Ross

[20 August 1999]

Hello,

Thanks again for your reply. No, I do not have any objections to use my stuff in any way, except that I would prefer that my e-mail address (if you plan to publish it) should be XXXXX and not my office address.

Sincerely