The following is the second in a series of interviews with Professor Antal E. Fekete.
The reader is encouraged to start here:
- A Conversation with Antal Fekete — Introduction
- A Conversation with Antal Fekete — Real Linear Algebra
- A Primer on Quotient Sets and Stepnumbers
Professor Fekete, here is a quote from the Symposia on the Foundations of Mathematics, held at Birkbeck College, University of London, on January 12-13, 2015.
“The focus of this conference is on different approaches to the foundations of mathematics. The interaction between set-theoretic and category-theoretic foundations has had significant philosophical impact, and represents a shift in attitudes towards the philosophy of mathematics. This conference will bring together leading scholars in these areas to showcase contemporary philosophical research on different approaches to the foundations of mathematics. To accomplish this, the conference has the following general aims and objectives. First, to bring to a wider philosophical audience the different approaches that one can take to the foundations of mathematics. Second, to elucidate the pressing issues of meaning and truth that turn on these different approaches. And third, to address philosophical questions concerning the need for a foundation of mathematics, and whether or not either of these approaches can provide the necessary foundation.”
What comes up for you as you read the above quote?
The organizers have missed the point of the conference. They must revise their program to include the antinomies (paradoxes) of mathematics and plans how to eliminate them.
Does each approach to the foundations of mathematics – set-theoretic, category-theoretic, etc., produce its own unique set of antinomies? Are there antinomies common to all approaches tried thus far?
These are the real questions I would like to see asked and answered by the organizers of the conference.
What do the antinomies in mathematics tell us? Are they simply nuisances? Curiosities? Or are they information-rich, indicating that the foundations of mathematics are unsound or worse?
They are hardly nuances or curiosities. They are indicative of deep-lying fractures in the foundations of mathematics that over a hundred years of intensive research has not been able to repair.
Professor, you suggest that set theory should be replaced with quotient set theory. In your estimation, what is the role of a quotient set-theoretic approach in building the foundations of mathematics?
The starting point of the quotient-theoretic approach is an upper bound, namely, the superset. The first antinomy ever popped up when people tried to attribute a meaning to notion of the “set of all sets”. It was not obvious at first that this notion did not make sense. When it dawned on people that it didn’t, the world of old mathematics was in ruins.
By contrast, there is no danger that the same thing would happen under the quotient-theoretic approach. It is immediately obvious that the notion “the quotient set of all quotient sets” does not make sense. As soon as you say “quotient set”, you have subjected it to an upper bound, the superset. This is one of the two main roles that a quotient set-theoretic approach to the foundation of mathematics is designed to play: a fixed superset exists. The other is ‘weak duality’ between the lattice of subsets and the lattice of quotient sets.
Could you spell out what this weak duality is and why it is important.
Let us imagine that we have a dictionary, with the aid of which we can translate sentences (definitions and theorems) from the language of quotient sets to the dual language of subsets. The duality is called ‘weak’ because its dual dictionary, with the aid of which we could translate sentences from the language of subsets to the language of quotient sets. For example, either distributive law is a valid sentence in the language of subsets, but the dual of neither is a valid sentence in the language of quotient sets.
Do you feel set theory supplanting subsets with quotient sets would eliminate many of the antinomies in modern mathematics? All of them?
Yes, certainly all those owing their origin to ‘inordinateness’. In the lattice of quotient sets the dual notion to that of “the set of all sets,” an inordinate notion, does not arise.
Professor, you write: “Early in my academic career I became fascinated with what Nicolas Bourbaki has, in the first book of his series Elements of Mathematics (first published in 1937), called a quotient set. In particular I was interested in the ‘weak’ duality between the theory of subsets and the theory of quotient sets.”
Why did weak duality hold your fascination? What information is weak duality trying to tell the world? Does it say that quotient sets are more fundamental that subsets?
No; it says just the opposite. The two, taken together, are equally fundamental. Duality is an organic part of the structure of our world. Without it there is no real understanding. Matter – anti-matter, injective – surjective; left-inverse − right-inverse; left cancellation – right cancellation; left-divisor – right-divisor; left-multiple – right-multiple; etc., these are just some of the dual entries in the dictionary mentioned above.
At the level of geometry this duality becomes even more interesting. Duality in projective geometry is no longer a weak, it is a strong duality. That makes projective geometry perfectly symmetrical. Characteristically, mathematics in the twentieth century caved in to the demands of fragmentation, the opposite of unification that duality is designed to promote. The elimination of projective geometry from the curriculum is one proof of that. The other is the failure to follow up on Bourbaki’s pioneering idea of a quotient set (that was dropped from the agenda later by Bourbaki himself). Nothing shows the prevailing confusion more convincingly than the lack of consensus in choosing a uniform notation for the ‘refinement’ relation in a quotient set: would it be ⊆ or ⊇ ? Half of the researchers chose one, the other half chose the other notation. Yet if we consider that the universal quotient set U is refined by every quotient set A (in other words, U is the most refined), in symbols, ⊆ , then it is abundantly clear that
A ⊆ B should mean that “A refines B”.
The controversy can also be settled by pointing out that the trivial quotient set 1 (that has just one element) is the coarsest (least refined) quotient set.
What are modern mathematicians missing about quotient sets that you feel you see clearly? Are the great still-unsolved problems in mathematics merely artifacts of the wrong approach to the foundations of mathematics? Would an expanded quotient set theory resolve these difficult problems?
This is a very good question. ‘Weak duality’ is crying to be recognized. And, lo and behold, even the discoverer of quotient sets, Bourbaki, fails to recognize it explicitly. He sacrifices the philosophy of mathematics on the altar of abstracting abstractions.
You write: “In enumerating stepnumbers we actually construct all finite quotient sets. Thus stepnumbers can be used to count the number of quotient sets. The great utility of stepnumbers is due mostly to this fact.”
The set of all finite quotient sets is a graded lattice, grading provided by the number of elements of the superset. The dual idea is that of the set of all finite subsets, which likewise is a graded lattice, grading provided by the number of all elements of the superset. If you think of it, in enumerating all natural numbers we actually construct all finite subsets. In this respect duality is complete.
Once it is pointed out that the binary number system uses the fewest digits least economically, it appears obvious that we are missing a number system at the far end of the spectrum that uses the most digits most economically. This missing number system is that of the stepnumbers. How is it that no one discovered the stepnumber system over a period of 300 years, since the inception of the binary number system by Leibniz in 1697? (Note that Fekete discovered stepnumbers in 1997.)
The reason for the oversight is that “most digits” mean “infinitely many digits”. People were not ready to accept the concept of a number system using infinitely many digits. They still aren’t.
I nearly fell out of my chair when you told me a few years ago that I was one of two people interested in stepnumbers! I would have thought you’d be inundated by the Department of Defense, Silicon Valley executives, quantum computing experts, cryptographers, number theorists of every spot and stripe… I could go on.
Actually, there were three. The other two were Rob Bechtel and his daughter, a high school student doing a special math course at Stanford University at the time. She was very talented. I am truly sorry that I have lost contact with her.
I empathize. I had the pleasure of crossing paths with a bright young woman, Katie, who is interested in clinical medicine, mathematics and computers, the combination of which, as I pointed out to her, quotient sets and stepnumbers are superbly suited to (as we shall discuss further). I also suggested that taking the initiative and becoming the world’s champion of stepnumbers — it’s there for the taking! — besides accelerating her understanding of mathematics and unifying her interests, might help her secure scholarships and/or grants — free money — for her education; why not get paid for doing what one loves? Medical school costs a king’s ransom, and non-dischargeable student loan debt is insidious because, aside from the horrendous long-term burden, in the event of misfortune it renders one a financial prisoner of the state for life. I was greatly looking forward to exploring quotient sets and stepnumbers together — (mathematics friends are hard to come by!) — and to introducing you two, thereby swelling our rank of stepnumberists, but alas … I too am truly sorry that I have lost contact with her.
‘And then there were two.’
Let us soldier onward. Certainly binary numbers have been profitable, figuratively and literally, theoretically and practically. And quotient set theory has already proven itself as a powerful and universal method to build the mathematics in general and statistics in particular that everyone uses every day at every level of society.
True. Quotient set theory is my proto-typical example of the “mad rush” to generalize in mathematics I have referred to in our first interview. Quotient groups, quotient rings, quotient spaces, etc., are all special cases of quotient sets that have been studied, possibly ‘over-studied’ by mathematicians without finding time to step back and ponder on the original idea of a quotient set with no encumbrance of a structure. Especially sore is the failure to study the lattice of finite quotient sets, the dual notion of the lattice of finite subsets, the bedrock on which the edifice of number theory, “the queen of mathematics”, has been built.
After a three century wait, if quotient sets and stepnumbers promise to be so powerful, why is there near-zero interest? One would think you’d get the Fields Medal in Mathematics (if they could get hold of you but for all the competing job offers).
In addition to the “mad rush” to generalize I can only think of the reluctance of the mind to work with a number system using infinitely many digits. It may appear as an “overkill” to people.
It’s ironic: our world is highly nonlinear and analog, yet we have a penchant for linear modeling with digital computers.
That much is understandable. It is natural that we are using linear approximations to describe non-linear phenomena, and digital computers instead of analog computers (soap bubbles). What is less understandable is our love affair with the binary number system, the first and simplest language for digital computers. We got stuck there and never inquired about possible improvements. If we had, the stepnumber system would have overtaken the binary a hundred years ago.
Is there a relationship between linear thinking and digital computing? Between non-linear thinking and analog computing? That is, do we gravitate toward linear models merely because 1) they tend to be all that we can solve analytically, and 2) because of the inherent limitations digital computing and binary numbers impose ? Or is there something deeper going on?
I think there is, namely stepnumbers. In fact, you cannot expect an answer to your question 2) before you gain more experience with the stepnumber system.
Representing numbers as binary numbers in digital computers in the general case requires truncation. And then, if these truncated numbers are operated upon iteratively, say in modeling with differential equations, the rounding error (never mind other errors, such as measurement error) — no matter how insignificant — roars to the fore even after a few iterations. Result: the model becomes worthless as noise will dominate.
Yes, truncation and rounding have been with us ever since we have been building mathematical models of the physical universe. All these models depend on a very slim mathematical base, the field of rational numbers. I think this slim base could be widened substantially through the introduction of the stepnumber system. I have reasonable hope that stepnumbers could be extended from integers not only to the field of rational numbers but also to the field of real numbers as well, although this hope is still at the conjectural stage. But if this hope materializes, tantalizing possibilities will abound, for example, there would be a much greater variety of periodic stepnumbers.
Will stepnumbers and analog computing – with the ability to represent and manipulate extremely large numbers without truncation – bring about a new golden era of modeling highly nonlinear systems?
Again, we are still at a conjectural stage. We have to be on our guard not to let our imagination run away.
How about possibilities for the theory of counting (involving combinations, permutations, and so on)?
The sky is the limit. Recall that the ‘milestones’ of the stepnumber system are the Bell numbers 1, 2, 5, 15, 52, 203,… (named after the American mathematician Eric Temple Bell, who wrote a paper on these numbers in the 1930’s), which count the number of quotient sets of a set of 1, 2, 3, 4, 5, 6,…, elements – dual idea to the ‘milestones’ of the binary system: 1, 2, 4, 8, 16, 32,…, (the powers of 2), which count the number of subsets of a set of 1, 2, 3, 4, 5, 6,… elements.
One of the representations of the Bell numbers bn in terms of infinite series (due to the Polish mathematician G. Dobinski who published his results in 1877, five years before Bell was even born) is:
bn = (1/e)( 0n/0! + 1n/1! + 2n/2! + 3n/3! + … )
Note the similarity of this series to the power series for Rényi numbers:
rn = (1/e)( 0n/0! – 1n/1! + 2n/2! – 3n/3! + … )
In exactly the same way we can have “counting numbers” exhibiting properties similar to those of Bell and Rényi numbers from the power series for sin x, cos x, … , sinh x, cosh x, …, etc.
That is, there exist integers an and bn such that:
0n/0! + 2n/2! + 4n/4! + … = ancosh(1) + bnsinh(1)
1n/1! + 3n/3! + 5n/5! + … = ansinh(1) + bncosh(1)
Furthermore, there exist integers cn and dn such that:
0n/0! – 2n/2! + 4n/4! – + … = cncos(1) – dnsin(1)
1n/1! – 3n/3! + 5n/5! – + … = cnsin(1) + dncos(1)
[For more on this subject see: Apropos Bell and Stirling Numbers, by Antal Fekete.]
Where do you see quotient sets, stepnumbers, and analog computing making the greatest impact?
The immediate impact will be on quantum computer technology that makes the binary number system obsolete (for reasons of jamming the memory units of digital computers with uneconomically long strings of binary numbers).
Perhaps we should characterize stepnumbers as “the world’s only sustainable number system”.
Continuing, you write: “Another application of quotient sets is epistemology, that part of philosophy which studies the nature and scope of knowledge. In all branches of science we have a hierarchy of concepts determined by generalization or specialization, respectively. This directly corresponds to the relation ‘refinement’ and ‘rarefaction’ of quotients sets. In other words, every branch of science can be characterized as a quotient set the elements of which are objects (or phenomena); and rarefaction is generalization while refinement is specialization.”
Indeed, this aspect of philosophy, that branches of science are just members of a hierarchy arranged in order of refinement or rarefaction, is sadly neglected. The best example I can offer is paleontology with its hierarchy of the geosphere, biosphere, protosphere, logosphere (also known as noösphere).
You also write: “Altogether there were four major upheavals during the evolution of the number concept. Each of them arose because of an obstruction, each being obstruction to the four inverse operations: subtraction, division, logarithm, and root extraction.”
Yes, this is another good example, concentrating on the evolution of epistemology of mathematics.
This is exactly how integers emerged from the natural numbers, overcoming the obstruction presented by subtraction; rational numbers from integers, overcoming the obstruction presented by division; real numbers from rational numbers, overcoming the obstruction presented by the logarithm operation and, finally, complex numbers from real numbers, overcoming the obstruction presented by root extraction.
[For more on the evolution of the number concept using quotient set construction to remove obstacles to inverse operations, see: Maiming the Mind, by Antal Fekete.]
Can we put these two ideas together and generalize and say that the quotient set construction can be used to remove obstructions to all sorts of mental operations and activities? In other words, do quotient sets represent the tools-of-choice to bring about future scientific revolutions, as they do in the case of revolutionizing the idea of a number system when introducing infinitely many digits culminating in the stepnumber system?
I firmly believe we can. It is necessary, though, to introduce an ‘inverse’ to the mental operation, such as ‘weak duality’, or ‘points at infinity’ in projective geometry.
You write: “Nature still uses form, rather than substance, for encoding purposes when it comes to genetics. Think of the DNA molecule and its duplication where it is not the substance that is important but the form, to wit: the double-helix arrangement. If it is true that the substance making up the human body is changed every seven years through metabolism, then, perhaps, our brain should be thought of, not as an aggregate of molecules, but as an arrangement of aggregates of molecules. Thus, then, quotient sets may be the appropriate tool to investigate the structure and operation of the human brain. What we are, our memory, our emotions, our psyche, are not encoded through the agency of substance. They are encoded through the agency of form. Substance is incidental and transient, and it can be substituted without changing the information content. More permanent than substance is form.”
The helix (spiral) is one of only three geometric forms in our three-dimensional space that can move within itself. The other two are: the straight line and the circle. Clearly, they can be thought of as a degenerate helix. This property of a curve being able to move within itself is paramount in solving the problem of reproduction, the problem of transmitting properties from one individual to another. I consider the double helix-idea as decisive in the age-old dispute whether form is superior and substance subordinate, or the other way round.
We cannot attach a date to the invention of the wheel. Although I haven’t investigated the question, I don’t think we can attach a date to the invention of the screw either (more precisely, the pair of the male and female screw).
[For an interesting computer application based on the differential geometry of the helix, see: The Greatest Slide Rule Ever Invented, by Antal Fekete.]
You also write: “Colin Blakemore, Professor of Neuroscience at Oxford University was asked to what scientific question he would like to have the answer during his lifetime most. His response was: ‘It would be nice to know how the human brain works.’ “
Someone should let professor Blakemore know about quotient sets.
Blakemore continues: ‘Indeed, presently we are nowhere near to answering this question. Everyone assumes that the human brain is some kind of computing machine. It receives data, processes it, and makes decisions. But what are the information-carrying agents, the zeros and ones of the ordinary computer, and what is the algorithmic process they undergo?’
I am afraid professor Blakemore is on the wrong track if he excludes stepnumbers and prefers to work with binary numbers in his search for information-carrying agents. And, since stepnumbers are merely numerical representations of quotient sets, the algorithmic processes the professor is trying to find are, possibly, just the superimposition and amalgamation of quotient sets (dual ideas to the union and intersection of subsets).
Blakemore concludes: ‘We just don’t know what the fundamental computing elements in the brain are. Is it a dendron; is it the spines of dendrites; or is it a synapse? We just don’t know.’
Again, professor Blakemore may be in danger of taking the wrong turn at the fork. His comments sound like a preference for substance over form. If dendrons, spines, dendrites and synapses are all substances and none of them is a form, then I’d venture to say that he will not live long enough to solve the problem how the brain works.
After the end of Blakemore’s quote you write: “In the absence of this knowledge [of what the fundamental computing elements of the brain are] the available mathematical models from computer science are of no use.”
Of no use! Are not the available models from computer science worse than being of no use? That is, do not the available models from computer science hold investigators back in their understanding of how the human brain works via what I would call the “tyranny of the model”? Binary numbers and digital computers are tightly woven into the very fabric of our existence − including how we think about thinking − so much so that they have become in our minds an emotionally potent oversimplification that leads investigators to chase the wrong model − at great loss in time, money, and opportunity cost − without ever questioning their underlying assumption as to whether the model is even remotely applicable. Your thoughts?
I could not improve on your formulation. It is interesting that you bring in emotions. Perhaps you should also bring in psychiatry.
Professor, you read my mind (pun intended). I think a crystal clear example of the extreme lack of real understanding of how the mind works, in part due to chasing a binary-based computer model of the mind − of chasing substance over form − manifests in the field of psychiatry. Currently psychiatry’s bible cataloging pseudo-diseases is the Diagnostic and Statistical Manual of Mental Disorders (DSM), the standard classification of “mental disorders” used by mental health professionals in the United States. In the words of Dr. Dilip Jeste (former President of the American Psychiatric Association), the “D” in DSM stands for “dart-board”. Even the U.S.’s National Institute for Mental Health (NIMH) has distanced itself from the DSM. That does not stop its being inflicted on millions of hapless victims. Once randomly “diagnosed”, patients are handed a life sentence as they are solemnly informed that they have a “chemical imbalance of the brain” (for which there is no evidence, let alone test) and are invariable given a “cocktail” (note the enticing marketing euphemism) of incredibly powerful and toxic psychoactive drugs. The patients are told that “everyone is different”, so it is “necessary” to try many different combinations of drugs − at great risk to the patient − to “get it right.” Then patients are told that their “custom-tailored” drug regimen will probably only be effective for a limited time, and that the game of darts will have to start all over … and over … and over. (If all of this is not an admission that psychiatrists have no clue, I don’t know what is.) Given the long list of available drugs, the number of possible combinations − and catastrophic effects (“side-effect” is another euphemism; there are only effects) − is enormous, yet the high priests of psychiatry operate as if they have a hotline to Heaven. They don’t; they are guessing. This is not to say that people do not suffer from severe problems; it is to say that something is seriously wrong with the model of how the brain works that serves as the basis for psychiatric treatment. And that model worships substance over form. The interested reader might watch The DSM: Psychiatry’s Deadliest Scam, and investigate the more enlightened works of Dr. Joanna Moncrieff, beginning with her lecture The Myth of the Chemical Cure: The Politics of Psychiatric Drugs. Think twice before becoming someone else’s lifelong financial annuity!
Professor, you further write: “There is one clue pointing to the possibility that quotient sets have a role to play in the working of the human brain, as distinct from the brain of animals. While animals, like humans, can conceive of a subset − for example, a monkey can distinguish between green and ripe bananas; only humans can conceive of a quotient set, that is, a complete classification with respect to an equivalence relation. For example, a monkey is unable to distinguish between a pile of ten bananas and a pile of eleven bananas. To do that, monkeys should be able to classify piles according as they have the same or different number n of bananas. This they are not able, nor can they be taught to do. The ability to handle quotient sets may be the very criterion that separates the working of the human brain from that of the animal brain.”
Are there other clues that quotient sets have a role to play in the working of the human brain, as distinct from the brains of other animals?
A reader has objected by saying that even the brain of a dog can handle quotient sets, according to Pavlov’s “conditional reflex” experiment. The ringing of bells can be interpreted by the dog’s brain as a quotient set in a spectrum that also embraces odors. The configuration of smells may be a quotient set of ‘monochromatic odors’.
I don’t know whether this objection deserves to be treated with respect, or it should be treated as a joke. Dogs, blessed as they are with a superb sense of smells, apparently failed to develop a theory of odors on the pattern of our theory of electromagnetic waves that embraces light, radio and t.v. signals, X-rays as well as cosmic rays.
Beyond our unique capacity for abstracting the number concept, can we not see quotient sets at work in man’s universal tendency, covering all of time and space, for expressing himself − indeed, for “knowing” the world around him − using models, analogies, similes, parables, hieroglyphs, letters, words, novels, plays, poems, drawings, paintings, cartoons, music, dance, and on and on? Should we not be relabeled as homo metaphora – the metaphor man?
You have a good point. Tell the Chinese, please, that they could improve on their system of writing by transliterating illogical characters using logical stepnumbers.
Can we not also see quotient sets at work in our deep sense that “all men are created equal.”
Only in contrast to the dictum “all mammals are not created equal.” All mammals belong to a quotient set, one class of which is that of men.
Winston Churchill wrote: “To each there comes in their lifetime a special moment when they are figuratively tapped on the shoulder and offered the chance to do a very special thing, unique to them and fitted to their talents. What a tragedy if that moment finds them unprepared or unqualified for that which could have been their finest hour.”
Churchill was only plagiarizing Shakespeare who said long before him that:
There is a tide in the affairs of men.
Which, taken at the flood, leads on to fortune;
Omitted, all the voyage of their life
Is bound in shallows and in miseries.
On such a full sea are we now afloat,
And we must take the current when it serves,
Or lose our ventures.
Julius Caesar, Act 4, scene 3
Indeed, it would have been more elegant had I quoted the Bard of Avon rather than Winnie the Pooh. But the point I was groping clumsily toward is that, for bright talented young people like our Miss Bechtel and Katie mentioned earlier, do not quotient set theory and stepnumbers present unique opportunities — special moments of a lifetime?
In my humble estimation, quotient sets and stepnumbers seem like a vast rich claim with not a footprint in sight. I imagine that they are subjects bursting with deep, interesting, productive projects for everyone ranging from seven-year-olds who enjoy puzzles and numbers and playing with steplogarithms using your Rainbow Slide Rule, to undergraduate and graduate students in search of meaningful thesis projects, on through to hardcore curmudgeonly mathematician, and beyond, to astute and nimble business men and women looking to make honest money by honestly serving humanity.
Modesty forbids me to comment.
Well, speaking as Max Photon, the brightest guy in the entire universe, I certainly respect the virtue of modesty.
Let me pose the question more tactfully. As a strong advocate for quotient sets, and as the discoverer/inventor of stepnumbers, do you have anything to say to our readers about “the chance to do a very special thing, unique to them and fitted to their talents”?
Yes. They should start by looking for examples of quotient sets around them. Seeing the connection between stepnumbers and finite quotient sets may be fitted only to the best talents.
Is there a pot of gold at the end of the rainbow?
Yes, and you can get it if you take a ride on the back of stepnumbers.
Before we end, are there any other thoughts you’d like to share about quotient sets and stepnumbers that I might not have touched upon, but that would help to illuminate the fruits of your remarkable efforts?
The teaching of mathematics would be revolutionized if we started in kindergarten, showing pupils examples of finite quotient sets (e.g., coloring balls, or color-coding them, which is an entirely different problem!) and counting-problems involving partitio numerorum e.g., in how many different ways can we change a $1 bill into coppers and nickels, dimes and quarters? Or: in how many different ways can we stick postage stamps for a total value of $1 along a line on an envelope using denominations 1-cent, 5-cent, 10-cent and 25-cent stamps? (Note that in the second example the number of solutions is greater because the order of the stamps matters, while the order of the coins doesn’t.)
These problems come from the same turf as the most intelligent toys, card games and board games do.
I also suggest that sighted pupils be taught Braille, and pupils with unimpaired hearing be taught sign language at a very early age (say, 2 and a half year).
I don’t know whether the Chinese have an equivalent of Braille, but if they do, an interesting question is on what principle is it based?
My last question is the best: can ultra-sound be utilized to make the blind see through a touch-pad?
Professor Fekete, this conversation has been fascinating and profound. Thank you so much for your time.
You are most welcome. I appreciate your interest.