We have shown the symbol √-a to be void of meaning, or rather self-contradictory and absurd.Augustus De Morgan (1806-1871), Professor of Mathematics, University College, London, 1831

The imaginary numbers are a wonderful flight of God's spirit; they are almost an amphibian between being and not being.

Gottfried Wilhelm Leibniz (1646-1716)

In the metaphysic of imaginary numbers, they possess, in terms of the distinction of Frege, a meaning of

sensebut noreference. Real numbers refer to features of the real world. Imaginary numbers do not. Thus, you can count5cows; but you cannot count5cows. The truth of imaginary numbers is in their beauty; but it is a beauty detached from phenomena, the way a flying airplane is detached from the ground. To some, the aviators, this will be more appealing.iἘγκλινοβάραγγος (Enklinobarangus)

Why the square? Probability is always positive, but waves generally oscillate between positive and negative values. But

the square of a number is always positive, and it is the square that is related to probability.Lee Smolin,

Einstein's Unfinished Revolution, The Search for What Lies Beyond the Quantum[Penguin Books, 2020, p.34] [note]

Too many people are being made to feel stupid for asking questions about math that sound basic but are actually important and profound. They have burning questions, but they're told those are stupid questions, or that they're not supposed to ask those questions in math, when really the questions are mathematically very interesting...

I just think it's a shame that so many people think they don't like math when they're only been shown

some very narrow, unimaginative, authoritarian versionof it, a version that doesn't allow for any personal input and curiosity of their own.Eugenia Cheng,

Is Math Real? How Simple Questions Lead to Mathematics' Deepest Truths[Basic Books, 2023, pp.3-4, boldface added]

Since this essay was originally posted, I have received comment at intervals about it, including a lengthly exchange that is posted under Correspondence. The predominant tone of responses seems to be surprise that anyone would worry about the legitimacy or meaning of imaginary numbers. Well, those concerned with the foundations of mathematics worry about the legitimacy and meaning of "**1+1=2**," not because anyone doubts the truth or usefulness of that equation but because it is the job of philosophy of mathematics and of logicians to worry about what any mathematical expression is about, and what makes it true. Indeed, what I find the most curious is that investigating "**1+1=2**" seems to be taken as unproblematic, while the topic of imaginary numbers occasions surprise, indignation, and even hostility. One might think there is some uneasiness or defensiveness involved.

That mathematics can be done without these foundational worries doesn't matter. The idea that *because* mathematics can be done without these worries, then they are stupid or worthless or meaningless may show a couple of possible things: (1) hostility to philosophical questions, or (2) ignorance of the degree to which foundational inquiries, with the likes of Cantor or Frege or Gödel, have contributed to the development of mathematics itself in the 20th century. Thus, the entire existence of axiomatic Set Theory derived from the questions about the foundations of mathematics. Indeed, foundational and metaphysical questions about mathematics are not actually part of mathematics, but of **meta-mathematics**, which has one foot in mathematics and the other in logic and philosophy. For the project here of the Friesian School, the development of meta-mathematics by David Hilbert (1862-1943) attracted the attention and entusiasm of Leonard Nelson, who realized that the principles of object language and meta-language had been introduced by Jakob Fries.

With the dismissal of philosophical questions or the refusal to see anything curious about imaginary numbers, I must say that such attitudes seem to me **complacent or deficient in curiosity** -- unless one frankly admits that they only find the mathematics as such, and not the philosophical meta-mathematics, interesting. There is nothing wrong with that. But when I see curiosity *decline* in proportion to the strangeness of a phenomenon, it strikes me as peculiar; and I should be forgiven for suspecting that some sort of uneasiness, defensiveness, or bad conscience is involved.

On April 25, 2019, I attended a talk at Villanova University by John W. Dawson, Jr., Professor of Mathematics, Emeritus, Pennsylvania State University at York. Professor Dawson catalogued the papers of Kurt Gödel at the Institute for Advanced Study and co-edited the *Collected Works* of Kurt Gödel. Dawson authored *Logical Dilemmas, The Life and Work of Kurt Gödel* [Routledge, 2005]. He is obviously one of the principal experts, if not the principal expert, on the work of Kurt Gödel.

Dawson's talk was about the relationship of mathematics to logic. His take was that, at the moment, it doesn't amount to much. While the axiomatization of geometry and the axiomatization of arithmetric, through Set Theory, are great achievements, other mathematics is done without reference or input from logic. Indeed, Aristotle's system of syllogistic logic, although antedating Euclid's *Elements*, has no obvious influence on it. Indeed, as Dawson noted, there are no syllogisms in the *Elements*. This should make us wonder about the nature of geometrical reasoning, and Dawson himself pointed out that the role of diagrams, although essential to all geometry, is not well explained by logic alone (once upon a time, sitting in the bar of the Texas Union, I tried to derive featues of conic sections from definitions, logic, and diagrams alone -- I didn't get far). The paradoxes of Set Theory and the dangers encountered of inconsistencies at the foundations of arithmetric tend not to trouble mathematicians in the least.

This should give us pause about the status of imaginary numbers. Arithmetic and geometry are clearly parts of different axiomatic systems; and the axioms of geometry are widely regarded as no more than empircal hypotheses, which are to be confirmed or falsified by observation or experiment (see A Deuteronomy of Kant's Geometry), and so, after a fashion, not even part of pure mathematics. The Continuum Hypothesis -- about the degree of Cantor's classification of infinites for the points on a segment of line, or real numbers with given limits -- which seems to be something like the *non plus ultra* for the logicians, stands as a possible axiom outside the axioms of Set Theory. Perhaps its reference to space makes it something more like an axiom of geometry.

So imaginary numbers are themselves axiomatically independent, with the inherent problem that they are founded on an original paradox. The very definitions of mathematical operations with negative numbers exclude them. The expresson "√-1" cannot be resolved with any algorithms for multiplication, division, roots, or anything else. It is a dead end, until squared.

Thus, Dawson's point is illuminating. As I have said, no mathematician needs to worry about meta-mathematical questions. The practice of arithmetic actually is untroubled by the potential dilemmas and paradoxes of Set Theory. Similarly, the use of imaginary numbers can happily skip over the paradox of their origin. But this does not make the meta-mathematical questions meaningless. And it is particularly curious to see actual hostility expressed, or concerns brushed off, by people who otherwise find Set Theory, with its history, problems, and lessons, unproblematic.

A principle that has been stated to me more than once to justify the reality of imaginary numbers is that a number is anything that is the solution to an equation. Since * i* is the solution to

There is no single number that solves the expression "**0/0**," as examined elsewhere. Zero is a rich source of paradoxes. The equation **5/x=y**, where **x=0**, means that **y** is going to be either infinite, which is not definite or very useful, or "undefined," which means that we've decided not to worry about it. If we say all solutions to **polynomial** equations are numbers, this rather stacks the deck, since polynomials are defined as only using the operations of addition, subtraction, multiplication, and non-negative integer exponents. How convenient. If we rule out any use of division, we aren't going to need to worry about division by zero. This would be the same, in effect, as to lay down the rule that division by zero will not be allowed in algebra -- which is more or less the case. By the same token, it could be laid down as a rule that taking the square root of a negative number will not be allowed in algebra: This is actually the practice of my calculator, which flashes "ERROR" when I try that operation. There is not such a rule, however, because the operation turns out to be very useful, and can even be rather well "defined," except that the definition looks merely tautologous, "The square root of minus one is the square root of minus one."

That is what I find the most exasperating about a particular attitude. If the expression √**-1** has a solution that is not a real number, what is that? Well, it is * i*. OK. But what then is

Again, it is important to recollect why these questions are being asked. If * i* is an unknown that is tautologously taken to be its own solution, does this mean that imaginary numbers are discredited and marked for banishment from mathematics? Certainly not. What imaginary numbers can do is not only essential to mathematics, with great elegance and beauty, but a great deal of fun and fascination as well, as considered below.

The principle that numbers are solutions to equations, or specific *kinds* of equations like polynomials, puts the cart before the horse. There are no equations without numbers, but there are numbers without equations. It is numbers that are foundational, not equations. It is numbers as such that must then first be justified, as indeed they are in Set Theory. That is the foundational project in mathematics, to build the system up from logically independent axioms, which means we go from integers to negatives, rationals, reals, etc. The actual *historical development* of whole numbers, zero, negative numbers, and then imaginaries is fully parallel to this, and not accidentally so. Putting equations, whose form develops along the way, *first* simply begs the question. Imaginaries result from certain *operations*, but these themselves must be defined first and their possible consequences (like division by zero) considered. There is nothing sacrosanct about them -- they do not self-evidently acquire a Midas Touch that makes everything in contact with them unproblematic.

The most troubling thing about the approaches I see are the implications of either *formalism* or *conventionalism*. "Formalism" means that mathematics is simply a symbolic system whose use and manipulations are independent of the meaning that any symbols may have. Thus, David Hilbert is supposed to have said that, "It must be possible to replace in all geometric statements the words *point, line, plane* by *table, chair, beer mug*." In those terms, we don't have to worry about what imaginary numbers mean because we don't have to worry about what anything means -- although Hilbert may have allowed, as Leonard Nelson certainly did, that the *axioms* of the system, from which the rest is formally derived, must themselves have meaning and real reference. For reasons detailed already below (the arguments of Gödel and Penrose), the idea that the *whole axiomatic system* could be without meaning or reference turns out to be quite false about mathematics.

"Conventionalism" is advocated by some historical mathematicians, like Poincaré, and it was consistent with most Mediaeval attitudes towards mathematics, which owed more to Aristotle than to Plato and his Pythagoreanism -- Platonism returned with Greek refugees during the Renaissance. Conventionalism (close to what is also meant by "Constructivism") means that mathematics is something that we've just made up and established conventions about and that therefore we can just do it, ultimately, any way that we like. Imaginaries are just one way we've decided to do it. Why we would then *bother* to do this can be handled with a pragmatic or "instrumentalist" approach, i.e. it gets us results that we want and so is justified because it "works" (apparently what even Stephen Hawking, sometimes, thinks about the entities of physics). Like formalism, conventionalism fails if mathematical truth depends at all on meaning and reference -- in other words, a truly formalistic or conventionalistic system would be based on its own internal principles and would require no *pied à terre* to anything outside itself. Such a requirement would involve a reference, which would compromise the conventionalism, and a meaning, giving the sense of the reference, which compromises the formalism.

While the value of formalism can be retrieved by attributing meaning to its axioms, conventionalism is in a deeper hole. The whole idea of a "convention" is that it involves an arbitrary choice, for which there is no reason or restraint. The conventionalist thus doesn't worry and doesn't think it even makes any sense to *care* about the meaning or reference of the convention. In relation to some mystical and quasi-religious conception like "truth" ("What is truth?" Pilate said), it just doesn't matter what our conventions are. Thus, if we imagine, for instance, that the rules for negatives in multiplication are simply conventions (a matter considered below), which could be done any number of different ways, we will not even ask the question *what* the meaning of this is and *why* it is to be done that way. If curiosity overcomes us, and we want to know why those rules are the way they are, we will set aside the convention and consider the reality.

I am sure that the appeal of formalistic, conventionalistic, or operative definitions of number is that treatments and definitions of number can be given, with a great deal of ink being spilling, without actually worrying about the truth or even the meaning of the matter. I expect that most of Number Theory today is simply a way of avoiding metaphysics, which mathematicians may be adivsed by Positivist or Wittgensteinian philosophers is meaningless. Sighs of relief. But the truth of mathematics, or its meaning, cannot be evaluated without metaphysical questions and answers. Truth, perhaps, we can do without; but Roger Penrose demonstrates that Gödel's Proof of the incompleteness of mathematics means that mathematical expressions must have meaning. So, without some attention to metaphysics, Number Theory is ultimately just irrelevant. And the popular notion that nihilism or materialism avoid metaphysical commitments is just confused.

Concern about the foundations of mathematics therefore means that the problem of imaginary numbers is in fact a *problem* and cannot be dismissed as superfluous or trivial. Indeed, if foundational questions can still be raised about "**1+1=2**," which they can (it is a delicate business in Set Theory to avoid paradoxes -- i.e. disastrously falsifying contradictions), it would be astonishing if imaginaries were somehow more obvious and certain. So, I'm sorry if some of you are determined not to worry about any of this. Don't bother reading the essay.

Another point I might make is about the nature of the metaphysics here. While my mathematical sympathies are Platonic, I am not in this respect a *strict* and proper Platonist, but a Kantian and a Friesian. A Kantian philosophy of mathematics is going to be *realistic* in relation to empirical and phenomenal reality but *not* in relation to things in themselves. It is, in Kant's terms, a case of "empirical realism" but not "transcendental realism." This can confuse people, but it is essential to the resolution of the matter here, which takes advantage of the difference by allowing that imaginaries can realistically exist in our mathematical representation, and in the phenomenal world, without having a sort of Platonic transcendent reference. It pains me to offend the mathematical Platonists, since in general they are the solid Realists in the metaphysical debates about mathematics, and I do think that the greater threat and error is from the conventionalists and formalists, but Kantian and Friesian modifications of Platonism are warranted.

All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.Leonhard Euler (1707-1783), 1770

The employment of the uninterpretable symbol √-1 the intermediate processes of trigonometry furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question.

George Boole (1815-1864),

Laws of Thought, 1854

**Isaac Asimov** (1920-1992) wrote an essay on imaginary numbers called "The Imaginary That Isn't" [printed in *Adding a Dimension*, Discus Books/Avon, 1975, pp. 68-79]. He began with an anecdote from his college days, when he overheard a friend's sociology professor claim that mathematicians belonged with poets and theologians as "mystics." Asking him why, the professor said that they believed "in numbers that have no reality," namely, the square root of minus one. Asimov protested that "the square root of minus one is just as real as any other number." *Pace* the departed and beloved Asimov (1920-1992), this is not quite right, but he certainly is correct that it does not turn mathematicians into mystics.

Imaginary numbers are an intriguing artifact of negative numbers. It is hard not to have negative numbers for many practical purposes. A debt is negative wealth. If a point is marked as zero anywhere on a line, and length is measured positively in one direction, then it is most conveniently measured negatively in the other. This also makes it possible to conceptually simplify some arithmetic operations: We no longer need subtraction as a separate operation from addition; instead subtraction is just the addition of negative numbers.

When negative numbers are multiplied, we are given certain *rules* for the operation: (1) When two positive numbers are multiplied, the result is positive; (2) when two negative numbers are multiplied, the result is also positive; and (3) when a negative and a positive number are multiplied, the result is negative. The only number that is neither positive nor negative is zero, and zero multiplied by anything (positive, negative, or zero) is zero [note].

One result of these rules is that when a number, positive or negative, is multiplied by itself ("squared"), the result will always be positive. This is what creates the difficulty. A number that is the actual result of an operation of squaring will always be positive, which means that the *square root* of that number can be either positive or negative. However, what if we have an equation which calls for a square root, and it just so happens that the value we want or need to put in is a **negative** number? Neither a positive number nor a negative number nor zero, when multiplied by itself, would produce a negative number. Thus, by *complete induction* (considering all possibilities), there is no number that is the square root of a negative number. Since any negative number can be factored into a positive number times minus one, the question can simply be stated as the problem of √**-1**: the number, conventionally represented by "* i*," that does not exist. This symbol was introduced by

If a number does not exist, the commonsense temptation is to say that it is meaningless and nonsensical and cannot be used. However, we have seen, by its definition, that √**-1** is certainly *not* meaningless. Even oxymorons are not meaningless. Even an expression like "square circle" is not really *meaningless*. The meaning may contain an inherent contradiction, which would make something of the sort **impossible** as an object, but we very definitely **understand** what the meaning is. Indeed, if we did not understand the meaning, then we would not be able to *evaluate* the contradiction. If we did not understand the meaning, then we would not *know* that it is an impossible object. We *know* from the meaning that a "square circle" is an impossible object. Similarly, we know from its meaning that √**-1** is an impossible object.

Asimov tries to handle this by ignoring the meaning. He says, "As long as some defined quantity can be made subject to rules of manipulation that do not contradict anything else in the mathematical system, the mathematian is happy. He doesn't really care what it 'means'." Then Asimov proceeds to model imaginary numbers with the "complex number plane," where one dimension is "real numbers" (positive and negative), the second dimension is "imaginary numbers" (positive * i* and negative

However, when Asimov says that a mathematician "doesn't really care what it 'means'," **this is not true**, and he should have known better. It is a principle of formal logic that the meaning of the terms doesn't matter, and the great mathematician **David Hilbert** expected that mathematics could be turned into such a logically formal system, where the terms could as easily have meanings like "beer stein" as much as their real mathematical meaning. This project, however, was sunk by **Kurt Gödel**, who famously demonstrated that any formal deductive system of mathematics must have statements in it that cannot be proven within the system be must nevertheless be true. **Roger Penrose** points out that what this means is that such statements must be true because of their **meaning**, since they cannot be made true by anything in the formal system. Hilbert's beer steins thus leave mathematics, and Asimov's appeal to the complex number plane, although a nice model and an intuitive aid, cannot be a substitute for or an escape from the actual meaning of imaginary numbers.

We can get a better idea of the meaning of imaginary numbers by examining some specific cases. At right we have, in blue, the graph of the equation in blue, which is a simple equation for an **ellipse** that has its center at the origin of a set of rectangular coordinates. At left we have, in blue again, the graph of the equation in blue, which is the corresponding simple equation for a **hyperbola** that has its center at the origin of the coordinates, like the ellipse. (These are schematic diagrams, not accurately plotted for specific values.) In each case we have a similar curious effect: Values of x and y that do not correspond to points on the curves yield imaginary results. Thus, for the ellipse, if we set **a** (the semi-major axis) equal to **4**, **b** (the semi-minor axis) equal to **3**, and **x** equal to **5**, then **y** comes out as **2.25 i**. This not a point on the ellipse, so the imaginary result tells us that a value of

But this does not mean that the imaginary results are meaningless or entirely without significance. If we work out many values of **x** and **y** for each graph, we get an interesting symmetrical effect: The graph for an real ellipse contains the graph for an imaginary hyperbola; and the graph for a real hyperbola contains a graph for an imaginary ellipse. The two graphs are mirror images of each other, in that what is imaginary in one is real in the other, indicated with the contrasting blue for the real curves and red for the imaginary curves. There is another interesting effect. In the ellipse, **a** and **b** represent half of the maximum (major) and minimum (minor) diameters of the figure. In the hyperbola, **a** is the "semi-transverse" axis, which, as in the ellipse, measures the distance from the origin of the coordinates to the point of the curve on the **x** axis. The transverse axis itself is then the minimum distance between the two limbs of the hyperbola. On the other hand, **b** in the hyperbola, the "semi-conjugate" axis, is not the obvious measure of anything. It actually measures the minimum diameter of the *imaginary ellipse* that goes with the real hyperbola. The conjugate axis of a hyperbola is, in a sense, an imaginary measurement.

If we take this seriously, it allows for a consolidation of the equations for an ellipse and a hyperbola: All we have to do is see a hyperbola as *an ellipse with an imaginary semi-minor axis*. Since **b** is squared, this will always give us a negative term in the equation, which is the *only difference* between the equations for the ellipse and the hyperbola. Also, since the transverse axis for a hyperbola is measured on the convex side of the curves, while the major axis of an ellipse is on the concave side of the curve, we might also consider a hyperbola to be an ellipse with a negative major axis (i.e. going in the opposite direction); but here, where this will always be squared into a positive number, this is not a significant possibility.

This, however, gives us a very real significance for imaginary numbers: If attempting to take the square root of a negative number produces a non-existent number, then squaring that non-existent number produces a real (negative) number again. Thus, unlike square circles, imaginary numbers can easily produce **real results** from the appropriate operations. An ellipse with an imaginary dimension may seem like an impossible object, but it isn't. It simply turns out to be a **real** hyperbola -- an ellipse turned inside out.

The objection that one may then have is that a non-existent object should not be able to become "real" through *any* "operation." As Parmenides would say, nothing can come from nothing. We know from life, however, that something impossible at one point in time may become possible later. What we need is a metaphysics of possibility and, as it happens, a metaphysics of non-existent objects. If imaginary numbers don't exist, how can they nevertheless exist? In fact, they exist in the same way as other non-existent objects, in the *representation of subjects*, i.e. mathematicians and the rest of us. One may object, "That doesn't count"; but this is a mistake, on the basis of several interpretations. First, if we adopt the metaphysics of unqualified Advaita Vedânta, the representation of a subject, which means *Brahman*, is *all that exists* -- "real" objects are simply *certain* objects that exist as representation. They are never "really" separate from Brahman.

This may be a bit too much in the way of metaphysics just for us to get imaginary numbers -- it might make it sound like mathematicians really must be "mystics" after all, as the professor of Asimov's anecdote said -- but there are other possibilities. The second is suggested by quantum mechanics, where reality, while "really" being out there, is at the same time conditioned by the observation of subjects. Observation collapses the sum of *possibilities* into discrete *actualities*. This gives to subjects a role in the constitution of reality that was always objectionable to a four-square Realist like Albert Einstein. The place of observation in quantum mechanics has stood up under every experimental test that has been devised to challenge it, including ones ultimately suggested by Einstein himself. But, however novel in science, this effect of the subject on objective reality is nothing new in philosophy: **Immanuel Kant** already saw *phenomenal objects* in experience as the result of an interaction between external and internal, which makes it possible to speak of a Kantian interpretation of quantum mechanics. The effect of this, which elevates the subject to something nearing an equal partner with objects in reality, is to dignify non-existent objects, such as imaginary numbers, with greater reality than they would seem to possess otherwise.

A third possible metaphysics for non-existent objects is just to posit *complete equality* between subject and object. This is the thesis of *Ontological Undecidability*. What it will mean is that when non-existent objects exist in the subject, in representation, they are in an ontologically significant realm of **possibility**. Whether they emerge into reality depends on a number of things. Square circles are impossible because they violate non-contradiction. If we think that all reality is absolutely limited by non-contradiction, then there will be no square circles. If we do not think that all reality is absolutely limited by non-contradiction, as **William of Ockham** thought was the case for God, then somewhere, somehow there may actually be square circles -- though this is certainly inconceivable for us. Imaginary numbers differ from square circles in that we can see how real results can be derived from them. That there actually are no imaginary numbers now does not mean that they are nothing: They exist conceptually, or intentionally, in representation, which is as much reality as they need before some operation produces real numbers and real results.

Imaginary numbers are indeed imaginary. Isaac Asimov need not have fled from that characterization [note]. But what is needed to make sense of it is the kind of metaphysics, of possibility and non-existent objects, suggested by quantum mechanics and completed by a Kantian theory. Mathematicians, indeed, don't need the metaphysics to do mathematics, so long as they can get by without worrying about what the constituents of their science **mean**. Much can be done in that mode, but it cannot be said, as Asimov wanted to, that it doesn't **matter** what it all means. It does, for mathematics itself, once the foundations of mathematics are at issue. And, once the question about meaning is seriously asked, it is not surprising, as in any question about meaning, that philosophy and metaphysics should be involved.

My interest in mathematics is, I am afraid, just casual enough that sometimes I realize that I've failed to notice the existence of a whole area of the science. This just happened with a book by John Stillwell, *Yearning for the Impossible, The Surprising Truths of Mathematics* [A.K. Peter, Ltd., Wellesley, Massachusetts, 2006]; and what I had never heard of was a part of mathematics relevant to the issue here, imaginary numbers.

...complex numbersa + bican be regarded asordered pairs of real numbers (a,b)....It was the Irish mathematician William Rowan Hamilton who in 1835 first suggested treating complex numbers as pairs of real numbers. The idea has merit in reducing properties of complex numbers to properties of reals -- and thus avoiding the mysterious √-1 -- but it does not tell or suggest anything new about complex numbers. [

op. cit.p.130]

Here we have a minor admission, that the √-1 is "mysterious" and perhaps something to be avoided. However, we have *already seen*, in effect, the move of understanding imaginaries as pairs of real numbers; for the use of the "complex number plane" is no less than to substitute something real, the plane, for the meaning of √-1, something which, as it happens, is governed by an ordered pair of *coordinates*. If one wants to argue that imaginaries are just as "real" as the reals, it is certainly an easy way to accomplish that if the imaginary factor can simply be eliminated and ignored. An honest way to do it is to overtly interpret complex numbers as "ordered pairs of real numbers (*a,b*)." A less honest way is to use a model, the complex number plane, that (covertly) accomplishes such a thing, without admitting (or perhaps realizing) that this is what it does.

This is not, however, the most interesting thing that Stillwell is introducing. For, as he says, "Hamilton's agenda was actually to find an arithmetic of *triples* (and after that, quadruples, quintuples, and so on). He hoped that his arithmetic of pairs would suggest a general rule for doing arithmetic on *n*-tuples for any positive integer *n*" [p. 130]. To develop his arithmetic of triples, i.e. ordered sets *(a,b,c)*, Hamilton expanded the complex number plane into a complex number *space*. As the second dimension was the imaginary axis *i*, Hamilton made the third dimension *another* imaginary axis *j*. So ** i^{2} = -1** and

Hamilton's arithmetic of triples didn't work; but it led him into something that did work, not an arithmetic of triples, instead an arthmetic of *quadruples*, or "quaternions," with ordered sets *(a,b,c,d)*. This then implied a *fourth dimension* to the complex number space, a dimension given by the product of *i* and *j*, to which Hamilton then assigned the variable *k*: ** ij = k**. Even better, this is another imaginary number. On 16 October 1843 Hamilton conceived his ultimate equality and wrote it on a bridge in Dublin:

Somehow this does not seem to have worked its way into mathematical lore quite like, for instance, Euler's Theorem:

Thus, ** ijk = ijij = -jiij = -ji^{2}j = jj = j^{2} = -1**. Other interesting relationships follow:

Stillwell goes on to illustrate the value of quaternions even in dealing with three dimensional space. But it certainly is an embarrassment of imaginaries: "...it is still a mystery what the product **ij** represents." Well, it represents √-1; but then, at this point, who the hell doesn't? It was bad enough when all Isaac Asimov needed to explain was *i*. What could he do with *j* and *k*? In this case, it would not be enough to assert that the "complex number space" explains it, because *k* is *not in* a three dimensional complex number space. The only justification for the multiplication of imaginaries, and the indroduction of a fourth dimension, is that it works. It didn't work for Hamilton's triples, but it did work for the quadruples.

The truth is that the imaginaries, by keeping the coefficients of the quaternions separate, and by occasionally multiplying out to -1, simply enable an arithmetic of **ordered quadruples of real numbers** to work. Thus we return to the original thought, that "complex numbers *a + bi* can be regarded as *ordered pairs of real numbers (a,b)*." So √-1 is simply a device of the apparatus, not a real number in its own right. As a number it is, to coin a phrase, "imaginary."

I find another aspect of this matter in the recent book *Not Even Wrong, the Failure of String Theory and the Search for Unity in Physical Law*, by Peter Woit [Basic Books, 2006].

The mathematics of two-dimensional surfaces is a beautiful and highly developed subject that goes back to Carl Friedrich Gauss in the early nineteenth century. This was investigated further by Bernhard Riemann and others later in the century, and their crucial insight is that it is a good idea to use complex numbers and to think of the surface as being parameterized not by two real numbers, but by one complex number. A surface parameterized in this way is called a Riemann surface. [p.116]

Where in Stillwell's discussion we might get around the peculiarity of imaginaries by seeing them as ordered pairs of real numbers, so that the complex number plane is, indeed, just a display of two real variables, now with the quote from Woit (a mathematician with a background in physics), we get something like the inverse of this. Since complex numbers can be modeled with a real plane, why not analyze real two-dimensional surfaces with complex numbers? Indeed, why not. This does mean, however, that the notion, as found in Asimov above, that the complex number plane is somehow the *real meaning* of complex numbers, is put in an odd position. If the complex number plane is the real meaning of complex numbers, now it would look like the real meaning of real planes is complex numbers! Riemann surfaces, however, simply reinforce the point made by Stillwell, that complex numbers can be viewed as ordered pairs of real numbers, which is why complex numbers can be matched up with real surfaces, from whichever *direction* one approaches the matter. It remains the case that √-1 is an operation for which there is no solution; and **"√-1 = i"** is a tautology, a stipulative definition, or perhaps an "Easter Egg" for -1.

Complex Numbers: The Misleading Story, by Fred Martin

Why I am a Platonist, Kantian Mathematics

Exchange with Correspondent on Calculus and Imaginary Numbers

Imaginary Powers of Napier's Constant

Lee Smolin is talking about the wave function in quantum mechanics. Whatever the physical meaning of the wave function -- and we get no clear answer from physicists -- everyone does agree that the square of the wave function tells us the *probability* of where a particle is to be found when the wave collapses, as when we go looking for the particle.

What is of interest for us on this page is Smolin's statement, "the square of a number is always positive." As we will see, most, if not all, mathematicians will be happy to chime in with, "No, the square of imaginary numbers is negative."

What is curious here is that Lee Smolin, as a physicist and not a mathematician, is that nevertheless he should know that there are both **real and imaginary parts** to the wave function. I have always wondered about the *physical meaning* of the imaginary part of the wave function, but then a lot of physicists don't worry about even the physical meaning of the *real* part of the wave function. So I've never seen an explanation.

Whatever Smolin thinks about that, what he does say creates a difficulty. Squaring the imaginary part of the wave function produces a negative number; but then, as Smolin says, "Probability is always positive." So what does it mean that, stricktly speaking, we end up wtih some negative probabilities?

Smolin, like so many, can ignore the physical meaning of the wave function; but he cannot ignore a result where we have negative probabilities. If that really means **nothing**, then why is there an imaginary part of the wave function in the first place? Discuss. I'll wait.

I don't think I've ever seen an explanation of the rules for multiplying positive and negative numbers. Usually we seem to just get the stipulation, "this is how we do it," or "this is how it works," which implies a perhaps arbitrary convention. If one's view of mathematics is conventionalistic, then this is sufficient. If, however, the development of numbers has some logical basis, more of an explanation is required.

Multiplication is really just an abbreviation of addition. A certain *number* (the "multiplier") of a particular number (the "multiplicand," i.e. Latin "to be multiplied") is taken and then all are added together. Thus, **2x3** means **3+3**. This adds up to **6** (the "product"). Now, as it happens, multiplication (usually) is a symmetrical relation, and we get the same results if we exchange the multiplier and the multiplicand. Thus, **3x2 = 2+2+2 = 6**. Because of this, it is typically unimportant to distinguish between the multiplier and the multiplicand. That is not the case in division, where it is not a symmetrical relation between the divisor and the dividend and we would get different answers if we exchanged those two. When we render multiplication into its equivalent addition, however, we can see that it is done differently and looks different depending on which is the multiplier and which is the multiplicand. The relation is no longer symmetrical.

Now, if the multiplicand is negative, we get the addition of a number of negative numbers, e.g. **3x-2 = -2 + -2 + -2 = -6**. This gives us the result we see in the rules, that multiplying a negative number by a positive number gives a negative number. It now becomes an interesting question how this works when the *multiplier* rather than the multiplicand is negative. If the muliplicand is positive, then we get a list of positive numbers, but since multiplication is symmetrical, we expect to get a negative product. How is that going to work? It will work, indeed, if a negative multiplier means that we ** substract** rather than add the numbers. It has already been observed that subtraction is the equivalent of adding negative numbers. So it is reasonable to expect that

It now becomes evident what happens when both multiplier and multiplicand are negative. We get the subtraction of a number of negative numbers, e.g. **-3x-2 = - -2 - -2 - -2**. All we need now is the principle that the negative of a negative is a positive, or the subtraction of a negative is the addition of a positive, which are not very mysterious (although certainly it is something, as Locke says, "whereof some Men think there may be a question made"). So **- -2 - -2 - -2 = +2 + 2 + 2 = 6**, which is the result from the rule that negatives multiplied by negatives are positive.

Indeed, I might consider the motivation for the equivalence, **- -2 = +2**. Why do we say that two negatives make a positive? The simplest analysis would be that **- -2 = 2 x -1 x -1**, where **- -2 = -2 x -1 = 2**, using the principles that multipying positives by negatives are negative and negatives by negatives are positive. If we interpret the **-2** as a negative number, and the added negative a substraction, I'm not sure how get from from there to a positive **2**. **-2** would need to be added to **+4** to get **+2**, and I don't see how that could be introduced. So using the multiplications may be the way to construct it.

What we needed for the rules about multiplying negatives was thus simply to distiguish the *operation* of the multiplier, addition or substraction, with the *number* of the multiplicand, positive or negative, together with basic rules like two negatives are a positive and the equivalence of subtraction with the addition of a negative. This all enables us to reduce the rules for multiplying negatives to basic operations of addition and subtraction.

The rule for multiplying two negatives can be illustrated geometrically. If we want to know what we get from **(a-b)(c-d)**, we can use a diagram. The Pythagoreans would have loved this.

At right, the area of the large rectangle is simply **ac**. To get the area of **(a-b)(c-d)**, all we need to do is substract the areas of the other parts of the figures from **ac**. So we start with **(a-b)(c-d) = ac-ad-bc**. However, we then notice that the area of **bd** has actually been subtracted twice. We can take care of that by adding it back once, getting us **(a-b)(c-d) = ac-ad-bc+bd**. This is what we expect from the standard way of multiplying **(a-b)(c-d)**. In that, multiplying the elements **(-b)(-d)** results in **+bd** as if we have been using the rule that multiplying two negatives results in a positive.

Taking **(a-b)(c-d) = ac-ad-bc+bd**, if we set **a = c = 0**, this results in **(-b)(-d) = +bd**, QED. This is what we wanted to find out. This demonstration avoids any of the rules for multiplying positives and negatives. All we have is addition and substraction, with multiplication used just to determine the areas of rectangles. While this does not explain things in terms of the nature of multiplication, as I have done above, it does give us something vivid and intuitive, like similar geometric demonstrations by the Pythagoreans.

We can derive other rules from the same equation. Thus, if **a=0** and **d=0**, then **(-b)(c) = -bc**. So a positive multiplied by a negative is a negative.

This proof is contained in part of a book that was kindly sent to me by Robert Naugle, *Elementary Mathematics from an Advanced Standpoint: Arithmetik, Algebra, Analysis*, Volume I of III, by Felix Klein (1849-1925) [translated from the German,
1932, p.26].

...I will state my own position dogmatically in order to avoid minor misapprehension. I believe that mathematical reality lies outside us, that our function is to discover orobserveit, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.G.H. (Godfrey Harold) Hardy (1877-1947),

A Mathematicians's Apology[Cambridge U. Press, 1940, 2019, pp.123-124]

In his book, *Negative Math, How Mathematical Rules Can Be Positively Bent* [Princeton U. Press, 2006], Alberto Martínez suggests that we might say that **-4 x -4 = -16**. If we could do this, of course, it would simply **eliminate** imaginary and complex numbers: **√-16 = -4** and **√-1 = -1**!

Martínez, however, seems to have a strong conventionalistic view of mathematics and would not view the rule that multiplying two negatives gets a postive is motivated by such an explanation as I have just given above. But this would all perhaps be shocking to Paul Nahin, who strongly endorses the "reality" of *i* -- and so, in his recent *Dr. Euler's Fabulous Formula, Cures Many Mathematical Ills* [Princeton, 2006], can only refer to the problem of *i* as the "**'**mystery**'** of √-1," with the "scare" quotes around "mystery." He quotes with approval a statement that, "Actually √-1 is a much simpler concept" than the irrationality of √2 [p.13]. Unfortunately, this would not explain why the Pythagoreans understood the latter but the former (or negatives, for that matter) was not heard of until many centuries later. Actually, √-1 is more "irrational" than the irrationals, and is so in a more troubling logical sense.

Martínez continues his conventionalistic ways in *The Cult of Pythagoras, Math and Myths* [University of Pittsburgh Press, 2012]. This book begins by debunking some myths about the history of mathematics, for instance that there is much in the way of evidence or reason to believe that most of what has been attributed to Pythagoras, including the Pythagorean Theorem, was or could have been accomplished by the historical Pythagoras of Samos. This gives sense to the title of the book and is a needed corrective to much of the ahistorical nonsense that is repeated about Pythagoras. However, as one reads on, the title assumes a different meaning, that the "cult" of Pythagoras is simply mathematical *Realism*, or Platonism. The impression that the use of "cult" may be intentionally disrespectful in this regard is reinforced at the beginning of the chapter "Inventing Mathematics," where Martínez says:

Through the so-called [?] Platonist outlook, many people construed mathematics in religious ways. They assumed [?] that its principles were eternal truths discovered by special men, geniuses [!], and they accepted that these truths were valid everywhere and could never change. The laws of Geometry and numbers seemed like the laws of God, and therefore mathematics was valued as a preparation to discipline the mind for studies of metaphysics and theology. In the 1730's, Bishop Berkeley [!] complained that some people accepted strange mathematical propositions on the basis of faith instead of reason. [p.181]

The tone of this passage seems contemptuous and dismissive, for which I would fault Martínez did I not feel the same way about quite a few modern philosophers, like Hegel. But this does conjure a scene-I-would-like-to-see of Martínez confronting Kurt Gödel, an undoubted "genius," whom Martínez himself quotes, in the only reference to Gödel in the whole book, as affirming his Platonism [p.40]. Since mathematical truths do tend to be discovered by geniuses, like Euclid, Fibonacci, Kepler, Descartes, Newton, Leibniz, Euler, Gauss, Cantor, Gödel, etc., but have been well understood by many others, I wonder what it is about genius that bugs Martínez.

Be that as it may, perhaps Martínez should be reminded that Bishop Berkeley did not believe that the visible world exists, and that Berkeley was the sort of extreme Empiricist who did not believe that points and lines, as defined in geometry, existed either. Martínez seems to endorse such a crude form of Empiricism. Thus, after Gödel, he might also want to confront Paul Dirac over the theory of "point particles." This all would be of a piece with Thomas Hobbes upbraiding Newton for the infinitesimals of calculus.

In teaching about mathematical realism, I used to draw a triangle on the blackboard to illustrate that geometry does not really talk about such concrete examples of triangles, whose lines are not one dimensional and cannot even be guaranteed to be straight (here the hypotenuse in this image is clearly a broken line of pixels). The properties of the triangle are in the mind, not on the blackboard. Martínez has a similar illustration, with the caption "This is not a triangle" -- rather like the Surrealist painting by René Magritte, "Ceci n'est pas une pipe" --

La Trahison des Images,"The Treachery of Images," 1928-9, René Magritte (1898-1967), Los Angeles County Museum of Art |
---|

This leaves me wondering if Martínez appreciates the meaning of the issue. He has already suggested that a "simpler way," than the proof in Euclid, in determining the ratio of the volume of a cone to a cylinder of equal base and height would be to physically build the cone and cylinder out of "paper" (which of course didn't exist in Euclid's day), fill them with water (which dissolves most paper), and compare the volumes [p.38]. This is "simpler"? That such an experiment would involve the *imprecision* that always attends such empirical procedures, in contrast to the precision of a geometrical proof, doesn't seem to bother Martínez. This might be the point where we reflect that he is actually a historian, not a mathematician. Of course, I'm not a mathematician either. And I have no objection to such constructions. I have made all of the Platonic and Archimedean Solids out of paper. But a paper construction isn't a proof. And much of the passage builds a straw man, with disparaging overtones about metaphysics and theology, as though Martínez "assumes" that we are aware of the bogus and empty nature of these disciplines, which would discredit historical mathematicians, all those "geniuses," if only we could reduce their judgment and motivation to that level.

But, to an extent, Martínez pulls back from the brink. He rejects various interpretations of mathematics, including formalism and constructivism, in favor of "pluralism," in which, together with pure conventions, "Some parts of mathematics represent physical patterns that exist independently of our imagination" [p.200]. While conceding some kind or degree of realism, this is revealing in its own right, since the reference to "physical patterns" is characteristic of the lack of recognition, appreciation, or perhaps even understanding of *abstract objects* on the part of Martínez. Quoting Berkeley may not have been incidental. Martínez's whole treatment of mathematics seems based on a very naive Empiricism or, at best, a view of mathematics as the equivalent of an empirical science as it would have been understood by a Logical Positivist. Points and lines? I don't see them. They have no "cash" value.

*The Cult of Pythagoras*, however, is not without value, because of the way Martínez tackles significant questions like division by zero, the multiplication of negative numbers, and quaternions -- matters that I have addressed in these pages. He asks far too many questions to just be a (competely) complacent conventionalist. He is indeed pleading a particular case and wants to get the conventionalistic lesson that he draws, but he is stirring things up that mathematical Realists might want to concern themselves with.

For instance, while division is the inverse function of multiplication (as subtraction is of addition, and integration of differentiation), it does raise its own characteristic issues. Thus, as **2x3 = 6**, we see the inverse function in **6/2 = 3**. One way we can understand this, or apply it in terms of "physical patterns," is to think of taking six items, like apples, and separating them into two equal groups. Each group will then contain **3** apples. We can also analyze the relation as **1/2 x 6 = 3**. Dividing by a number is the same as multiplying by the reciprocal of the number, just as we can read "one half of six." Multiplication by reciprocals is rather like analyzing substraction as the addition of negative numbers.

The fun starts with negative numbers. If **-2x-3 = 6**, then **6/-2 = -3**. But what does it *mean* to divide six items into *negative two* equal groups? I really cannot capture a physical meaning for this, and it certainly looks like no more than an *artifact* of the rules that we have accepted for multiplication. Perhaps like the way *imaginary numbers* are artifacts of equations that otherwise seem meaningful and useful. Since division by a negative number usually generates no problems, we could be good conventionalists and not worry about the matter. However, it does sometimes generate problems and paradoxes.

First of all, there is the paradox of the equation **-1/1 = 1/-1**. Here we have a quantity that is equal to its own reciprocal. That is unusual, although it looks a whole lot like a related equation, **1/1 = 1/1**. But as *numbers* minus one and one are separated by two whole numbers, as much as one and three. But the equation **3/1 = 1/3** is obviously false.

The peculiarities of **6/-2 = -3** and **1/-1 = -1/1** are each due to the same thing: A negative sign in the **divisor** can be *transposed* as a negative sign in the **dividend**. We can see this simply by multiplying the offending expressions by a number that is the equivalent of one, namely **-1/-1**. This gets us **6/-2 x -1/-1 = -6/2** and **1/-1 x -1/-1 = -1/1**. Thus, although division is not symmetrical, and we cannot commute the divisor and divident as we could the multiplier and multiplicand, we actually can commute the *signs*. Since we can always do that, we can always make a positive division of negative quantities, whether they are debts, red ink on business ledgers, temperatures below zero, depths of mines, longitudes West of Greenwich (if we do it that way), etc. If we multiply both sides of the expression **-1/1 = 1/-1** by **-1** (equals multiplied by equals are equal), as **-1/1 x -1 = 1/-1 x -1**, this is then **(-1) ^{2} = 1**, and

Trouble brews when we do something a little different. It seems like it would be unobjectionable to take the square root of each side of **-1/1 = 1/-1** (the square roots of equals should be equals), giving **√-1/1 = √1/-1**. So we get **(√-1)/1 = 1/√-1**, or ** i/1 = 1/i**. We can multiply that out to

What has gone wrong? Well, we stumble across a unique and peculiar property of ** i**. Where the number

Perhaps this is the smoking gun, the *reductio ad absurdum* and falsification of imaginary numbers. But since we can always transpose a negation from a divisor to a dividend, we can avoid the contradiction -- just as we avoid division by zero -- so perhaps this is just the *reductio ad absurdum* of division by negatives. But it is only a problem when we take the square root of a negative divisor. It is not too difficult to avoid that. We don't *need* to take the square root of a negative divisor before getting the negation out of it.

Since it may not have been obvious what division by a negative quantity was supposed to mean in the first place, perhaps this trouble with ** i** does no more than expose its meaninglessness. It should be a sobering moment for uncritical enthusiasts of imaginary numbers. It should also be a sobering moment for conventionalists. We can't just make mathematics work any way we want to. The square root of

As it happens, there is a sense in which *no* mathematician really believes that imaginaries are just as real as real numbers. I find this in the case of prime numbers. No mathematician ever hestitates to begin listing the prime numbers in this way: **2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,** etc.

However, if imaginaries are just as real as reals, there are many numbers in that list already that can be factored and are divisible by something besides themselves and one. Thus **5 = (2 + i)(2 - i)** -- 5 is the product of two

For instance, **89 = 4x22 + 1**. This works because, as was proven by Pierre Fermat (1607-1665), any prime that equals **4n + 1** is the sum of two squares: **89 = 8 ^{2} + 5^{2}**, where the numbers squared (

There is a way in which this is taken seriously, and a way in which it isn't. In the former respect, the notion of prime numbers is generalized into "Gaussian Primes" in which the characterisitics of complex numbers are encompassed. But, on the other hand, such complications can be ignored and the traditional study of natural number primes can just be continued. I think the latter is actually what continues to be the principal focus of interest in primes. Prime numbers are about the natural integers, which are all real. But this does give a priority to integers, and reals, that is implicitly denied by the argument for imaginaries as "as real as the reals."

Fermat also proved that any prime of the form **4n + 1** also is part of a Pythagorean Triplet.

The exception to algebraic commutation creates some problems for determining values involving ** i, j, & k**. Thus, if we begin with

Note that the reciprocal of ** j**, like that of

In any case, the result we get is both ** ik = -j** and

I am now informed by Martin Ondracek, at the Institute of Physics of the Academy of Sciences of the Czech Republic, about the convention employed to avoid the contradictions just described. He says, "either multiply both sides of the equation from the left (insert "i" to the front of the expression) or multiply both side[s] of the equation from the right (append "i" to the end of the expression)." This also means, "you never get to cancel something in the middle of a product, as you did in ijk/j." If this is enough to preserve consistency, then the convention would be logically justified (*salve veritate*). He also mentions, "this is the same with matrix multiplication, by the way."

So if we wish to multiply ** ijk = -1** by

There is an interesting aside from Dr. Ondracek:

It surprises me (and even bothers a little bit, as it is not what I would expect from a philosopher) that you were not curious enough to consult some mathematician to find out what mathematics has to say about this equation and what answer it provides.

While I am always "curious enough" to consult mathematicians, I don't know any at the moment; and my experience with inquiries to academics I do not know, including physicists and mathematicians, is often to get a brush off rather than an answer. The questions I ask often seem to be annoying, perhaps because I am curious about things that they are not. With pages on the Internet, however, which express precisely my own curiosity, I can always hope that they attract the attention of someone like Dr. Ondracek, who can give a succinct, *ad rem* answer to questions I have. He has done this. Even Dr. Ondracek, however, cautions that he does not speak for his institution. Perhaps the Czech Academy of Sciences would not want to endorse restrictions on the manipulation of quaternions.

He also seems to miss the point of this imaginary numbers web page, saying, "if I understand it correctly that Godel's incompleteness theorem might be somehow used to argue that real numbers are metaphysically real, then why not complex numbers by the same token?" Of course, it is Roger Penrose who argues that Gödel's Proof implies that mathematical expressions have meaning and reference -- *q.v.*. Why that argument would not work in the same way for imaginaries depends on the foundational question what the derivation and basis is of such numbers. That mathematics in general has meaning and reference does not necessarily mean that imaginaries, for all their meaning, also have reference. "The same token" is not logically transitive there. If Dr. Ondracek has no curiosity about *that*, then, as I say above, "So, I'm sorry if some of you are determined not to worry about any of this. Don't bother reading the essay."

There is a curious feature to the violation of commutation where ** ij = -ji**. As Dr. Ondracek notes, this is also characteristic of matrix multiplication. Now,

**Wolfgang Pauli** derived a physical attribute of great significance from this curiosity. It turns out, the violation of commutation is the mathematically basis of the **Pauli Exclusion Principle**, which is that more than one particle with half-integer spin cannot possess the same quantum numbers in the same physical system, such as an atom. The violation of commutation violates the **Bose-Einstein Statistics**, in which commutation is allowed but never makes any physical difference because of the absolute identity of identical particles. Now, only particles with integer spin, or "bosons," obey the Bose-Einstein Statistics, while the other particles with half-integer spin obey the **Fermi-Dirac Statistics** and consequently are called "fermions." The structure of the chemical elements and ordinary matter absolutely depends on the difference between bosons and fermions; for, if electrons were bosons, in the atoms they would all collapse to the lowest state of energy and angular momentum, making all atoms chemically identical.

The diagram illustrates the Bose-Einstein Statistics in the way they were explained by Albert Einstein himself in a 1925 letter to Irwin Schrödinger [cf. A. Douglas Stone, *Einstein and the Quantum, The Quest of the Valiant Swabian*, Princeton, 2013, p.239]. Ordinarily, if we toss two coins in the air, half the time (2/4) we will get one coin heads and the other tails. We get this result in two out of four tosses because one coin is one way and the other the other, but which is which can be switched. Now, Einstein decided that the "switched" cases are not going to be different in quantum mechanics because the identity of the particles means, not only is there no way to distinguish them, but that in principle they cannot be reckoned as distinct. The cases where one is one way and the other is the other are consequently the *same case* as far as quantum mechanics is concerned. The means that the probability of tossing the coins where one is heads and one is tails only happens *one out of three times* rather than *two out of four times*. This extraordinary identity of identical particles is only broken for fermions by the extraordinary violation of commutation that we see in matrices and quaternions. On the other hand, the mixing of identities, where it looks like one can be either heads or tails, as long as the other is the opposite, is an example of the mixing or "superposition" of particles, which is the result of adding their wave functions.