Realism, irrationality,

and spinor spaces

Adrian Heathcote

Abstract
Mathematics, as Eugene Wigner noted, is unreasonably effective
in physics. The argument of this paper is that the disproportionate
attention that philosophers have paid to discrete structures such as
the natural numbers, for which a nominalist construction may be
possible, has deprived us of the best argument for platonism, which
lies in continuous structures—in fields and their derived algebras,

Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce)

such as Clifford algebras. The argument that Wigner was making is
best made with respect to such structures—in a loose sense, with
respect to geometry rather than arithmetic. The purpose of the present
paper is to make this connection between mathematical realism and
geometrical entities. It thus constitutes an argument against formalism,
for which mathematics is merely a game with humanly set rules; and
nominalism, in which whatever mathematics is used is eliminable in

No 75 (2023), pp. 15–57 ∙ CC-BY-NC-ND 4.0

the final analysis, by often insufficiently specified means. The hope
is that light may be cast on the stubborn mysteries of the nature of
quantum mechanics and its mathematical formulation, with particular
reference to spinor representations—as they have been developed by
Andrej Trautman. Thus, according to our argument, QM may appear
more natural, as we have better reasons to take spinor structures as
irreducibly real, a view consonant with the work of Trautman and
Penrose in particular.
16 Adrian Heathcote

Keywords
indispensibility, nominalism, spinors, complex numbers, incommen-
surability.

any who have more than a passing interest in mathematical

M physics have been impressed by the intimate connection that
exists between quite advanced mathematics and the elucidation of
our best physical theories, and being so impressed have taken this
as an argument for a form of mathematical platonism. Yet, in the
wider philosophical community, and certainly in the culture at large,
nominalism seems (perhaps only to a jaundiced eye) to dominate. Thus
we have a rather stark opposition between philosophy and science in
which the two sides appear to be largely talking past one another, and
little that is said advances the debate in a successful manner,
The present paper is an attempt to get beyond this impasse by
offering a way of recasting the issues, so that 1) a central part of
the nominalist intuition can be seen to have some plausibility; and
2) that nevertheless the platonist can be seen to be correct in that
mathematical physics does in fact offer an argument for the reality
of mathematical entities. Indeed, my suggestion will be that there is
a straight line between the motivation for platonism among the ancient
Greeks and platonism today. Thus the main claim of the present work
is that there is a mechanism for the expansion of our mathematical
ontology that is directly tied to our progress in mathematical physics,
a connection that is unlikely to be accidental. In brief: the taking of
roots is often ontologically ampliative.
We may begin by noting that perhaps the most important way
that the discussion has gone astray is through the historical focus on
the arithmetic of the natural numbers, a focus that was present in
Kant as well as Frege, and that flowed naturally through the reductive
Realism, irrationality, and spinor spaces 17

programmes of the 20th Century. The natural numbers were seen

to have first place in the ordo cognoscendi: they were our original
mathematics—account for these and all else will somehow surely fall
into place. In due course philosophical discussion became bound to
the twin poles of arithmetic and set theory—the latter having first
place in the ordo essendi. Though nominalists and realists disagreed
on what should be in our ontology, they were at least disposed to
agree on what mathematics we should be considering.
The implicit thought here seems to be that whatever we can say
about the natural numbers we will be able to say about any other
mathematical structure. However I want to suggest that this is false:
that the natural numbers are a special case that lend themselves to
a very special nominalist explanation, an explanation that does not
extend to other mathematical entities in which we might be interested.

1. A nominalism for arithmetic

Let us begin by giving the Peano axioms in their second-order form.

We modify them in a way that is now customary by taking the first
number as 0. Since 0 is the additive unit it means that much more of
what would ordinarily be considered elementary arithmetic is deriv-
able. However it also means that we would have to be careful in
the statement of divisibility. Peano’s own statement would lead to
problems unless modified, for it would allow division by zero.1

1An extra condition stating that in all cases of 𝑚/𝑛, 𝑛 ̸= 0 would be sufficient. This
original axiomatisation is weaker than that of Hilbert and Bernays in their Grundlagen.
 18 Adrian Heathcote

Where Peano speaks, in the first axiom (and then throughout) of

the 𝑛 being a member of a set 𝑁 , I will be explicit that this set is to
be the set of natural numbers N.

Axioms for Peano Arithmetic

PI : 0 is a natural number;
PII : For every natural number 𝑛, 𝑛 + 1 is a natural number
PIII : For every natural number 𝑛, 𝑛 + 1 ̸= 0;
PIV : For all natural numbers n and m, 𝑛 + 1 = 𝑚 + 1 if and only if
𝑛 = 𝑚;
PV : If 𝜑 is a property of numbers such that: 0 is 𝜑, and for every
natural number 𝑛, if 𝑛 is 𝜑, then 𝑛 + 1 is 𝜑, then all natural
numbers 𝑛 are 𝜑;
PVI : 𝑛 + 0 = 𝑛;
PVII : 𝑛 + (𝑚 + 1) = (𝑛 + 𝑚) + 1;
PVIII : 𝑛.0 = 0;
PIX : 𝑛.(𝑚 + 1) = (𝑛.𝑚) + 𝑛;
PX : 𝑛.(𝑚 + 𝑝) = (𝑛.𝑚) + (𝑛.𝑝).

These axioms, as is well known, are derived from Dedekind’s

Was Sind und was sollen die Zahlen? (1888), and Dedekind had there
shown that his axiom-set is categorical. His method, as outlined in
his letter to Hans Keferstein in 1890, is not to appeal to known fea-
tures of the natural numbers—this, he says, would result in a vicious
circularity—but to give axioms that ought to determine any infinite,
well-ordered set (Van Heijenoort, 1967).
But now we come to the crucial point. Not only are these axioms
such that they characterise the natural numbers, they also characterise
the numerals that name the natural numbers. For the numerals are
 Realism, irrationality, and spinor spaces 19

also a well-ordered infinite set and begin with a first numeral ‘0’. To
achieve this isomorphism we must understand that numerals are not
identical with inscriptions of numerals: there are numerals that no one
will ever, or could ever, write down. But no matter, these numerals
exist and there are many that cannot be written down that can be char-
acterised by a definite description—thus the name “Graham’s Number”
is an abbreviation of a definite description where the numeral itself
could not be written down without a secondary abbreviated notation.
Of course, there will be some nominalists for whom an infinite set
of numerals is already going too far in the direction of platonism: it
must be understood that the way out of the problem that I am offering
here will not be a way that is open to them. But a rigid Inscriptionism
is, I believe, a most difficult position to extract explanatory content
from, and so we must await someone who is prepared to try to make
it work. At any rate I say no more about such a view here.
Allowing ourselves an infinite set of numerals we can check the
Peano axioms to see what they mean when applied to numerals. As
already noted neither Peano nor Dedekind mention numbers, for their
purpose in providing an axiomatisation is to characterise numbers
without circular descriptions. So, adapting Peano, we have simply:

P*I :0 ∈ 𝑁;
P*II :If 𝑛 ∈ 𝑁 then 𝑛 + 1 ∈ 𝑁 ;
P*III :If 𝑛 ∈ 𝑁 then 𝑛 + 1 ̸= 0;
P*IV : For 𝑛 and 𝑚 ∈ 𝑁 , 𝑛 + 1 = 𝑚 + 1 if and only if 𝑛 = 𝑚;
P*V : If 𝜑 is a property of the members of N such that: ‘0’ is 𝜑, and
for every 𝑛 ∈ 𝑁 , if 𝑛 is 𝜑, then 𝑛 + 1 is 𝜑, then all 𝑛 ∈ 𝑁 are
𝜑;
P*VI : etc.
20 Adrian Heathcote

Since addition is simply an operation that takes a member of 𝑁 to

another member of 𝑁 it is also well-defined on numerals: it is simply
counting forward. Likewise for multiplication. Thus the remaining
Peano axioms will also have a clear meaning.
Now the philosophical point should be clear: since there is an
isomorphism between the two models of the Peano axioms, and since
we use the numerals to speak of numbers, there is always a danger
that we will confuse the two—and, the nominalist may say, we have
confused them, and confused them throughout history. Thus we are,
whether we are nominalists or realists, simply creating confusion if we
say that ‘numbers can be written down’. I can write down a numeral
but I cannot write down a number. By analogy, to make the point clear,
I cannot write down Mary but I can write down Mary’s name, ‘Mary’.
So when we speak of writing down numbers we are already confusing
a name with the referent of the name. Thus in Peano’s axiomatization
what is written down and axiomatised are numerals.2
Now if we take a Medieval conception of nominalism, we may
hold that there are nothing but numerals, that these do not refer to
numbers, as a name refers to a thing, but that they are all there is
to what we think of as number. Thus numerals are a flatus vocis, in
Roscelin’s phrase, an empty wind, and mathematics is simply a game
with rules for the manipulation of these numerals. In the 19th Century

2 See Button and Walsh (2018) for a discussion of the rôle of language in axioma-
tisations, including second-order axiomatisations of arithmetic. In their section 1.13
there is again signs of confusion between numbers and numerals. Properly, however,
in such second-order axiomatisations we are quantifying over properties of numbers
themselves, but we then also sacrifice Dedekind’s desideratum of non-circularity.
Realism, irrationality, and spinor spaces 21

there is evidence that this was the view of Helmholtz and Kronecker,
though undoubtedly many others followed in the 20th Century, notably
the Formalists.3
Some credence is given to this position if we ask ourselves, sym-
pathetic to this nominalism, what the law of the commutativity for
multiplication means: if we multiply together two numbers a and
b then the order of the multiplication does not matter. But, says my
imaginary nominalist, surely the order of an operation suggests some-
thing that we do, some way of manipulating objects, in a particular
sequence, and the only objects available for us to manipulate are nu-
merals. Likewise with associativity: the order in which an operation
is performed suggests an action with consequences. After all, to add
and to multiply are verbs and require objects on which the action is to
be performed.4
Now I will say that I think we have here the beginning of an
interesting discussion about nominalism that could be developed
further, and one that would be helpful in clearing our minds of long
standing confusion. In particular it may help us understand what
we mean when we make a distinction between the potential infinite
and the actual infinite, for there is a clear sense in which there are
a potential infinity of numerals that we may write down. By contrast
I am not sure that sense can be made of saying that numbers themselves
are potentially infinite: either they are finite or they are infinite, and
there is nothing in between these two cardinalities. Nor, if it is numbers
themselves that are being thought of as potentially infinite, is it all

3 No direct evidence of Roscelin’s position survives, only the replies of his opponents,
such as John of Salisbury. Thus see Joseph Owens (1982).
4 One can find something of this view expressed in Whitehead’s Universal Algebra,

where he speaks in the introduction of 𝑎 + 𝑏 and 𝑏 + 𝑎 ‘directing different thoughts’.

I do not say that this performative interpretation of arithmetical operations is correct,
merely that if we have it then it seems most apt to apply it to numerals.
22 Adrian Heathcote

clear what would be releasing or realising this potential. For whom is

this potential realised? When is it being realised? Can these numbers
return to being unrealised? Confusion between name and referent is
rife in this area, and of long standing.
But I cut this discussion short to say that, ultimately, I do not be-
lieve that it can be correct for anything more than the natural numbers
(and in the light of an argument to come in §5, not even there). It
depends on our having numerals which can stand in proxy for natural
numbers and thinks of numerical operations as manipulations of those
numerals. But, as Hilbert realised, this cannot be extended to the real
or complex numbers—a point I come to in the next two sections.
However, I think that something like the above reasoning was
present to the Pythagoreans and Plato: as long as we had to think
only of the natural numbers we were able to be lulled into a state of
Nominalism about numbers. But when irrational magnitudes were
discovered there was no longer a way to avoid realism. The argument
for this, with some historical evidence, is given in the next section.

2. Plato and incommensurability

Mathematics began as an abstract discipline, I suggest—as opposed

to a pragmatic aid to accounting—with the Pythagorean discovery
that the square root of two cannot be either a whole number or a ratio
of whole numbers. There are now many proofs of this, but here is
a beautiful, little-known one by Theodor Estermann (1975). (It isn’t
known what proof the Pythagoreans actually used, though there has
been much speculation. Nor can it be certain that the Pythagoreans
were the first to construct such a proof.)
Realism, irrationality, and spinor spaces 23
√
If 2 were a fraction then there would be a set of natural num-
bers S, whose members when multiplied by said fraction would yield
a natural number. And if there is such a set then by well-ordering there
√
is a least member of that set: call it k . So k 2 is a natural number
and by definition the smallest such number. But on the hypothesis that
√
2 is a fraction we can find a number m that is smaller than k for
√ √
which m 2 is also a whole number. Thus consider m = k ( 2 −1)
√
= k 2 − k . We now have
√ √ √ √
𝑚 2 = (𝑘 2 − 𝑘) 2 = 2𝑘 − 𝑘 2.
√
This shows that m is a member of S since 2k − k 2 is obviously
a whole number. But this m is also less than k (the number 1 was
chosen specifically so that we would have
√
0< 2−1<1
√
and thus m = k ( 2 − 1) is less than k ). So we have found an m < k,
with m ∈ S , contrary to the hypothesis that k was the least member
in S . Repeating the proof will produce an infinitely descending set of
natural numbers, which is impossible.
The beauty of this proof, besides its great simplicity, is that it
relies only on the properties of natural numbers and ratios of same. As
Man-Keung Siu has pointed out there is an interpretation of this proof
in the geometry of triangles, but the proof itself is free of any geo-
metric assumptions.5 The proof can also be generalised to the square
root of any number that is not a perfect square, as Estermann noted,
while requiring no heavy theorems like the Fundamental Theorem of
Arithmetic.

5 Man-Keung Siu (1998). See also P. Shiu (1999).

24 Adrian Heathcote

The Pythagoreans of the 6th Century bc probably did not have

available this proof (if they had, the generalisations to other non-
square numbers would have been evident to them) but no matter—they
√
had some other that proved the same fact: 2 cannot be either a whole
number or a ratio of whole numbers. And it is a simple application
of the Pythagorean Theorem that the diagonal of a unit square has
√
a length that is 2 and so, such a length must exist. It was left to the
mathematician Theodorus to extend the proof up to 17 and Theaetetus
to generalise the discovery to the square roots of all numbers that are
not perfect squares (and again, we cannot be sure what proof was
used). By the time of Euclid this discovery was well-developed as the
theory of incommensurable magnitudes, and developed in books V,
IX and X of the Elements. In Book X Euclid extends the theory of
irrationals to all that have the form
√︁
√ √
𝑎 ± 𝑏.

A lost book of Apollonius is meant to have gone further and consid-

ered those that were unordered—possibly including 𝜋 .
The mathematical significance of this discovery has been thor-
oughly researched, by Knorr (1975) and Fowler (1999). But what
about the metaphysical significance? In metaphysical terms, what can
√
2 be, and what can it not be?
√
The best way to approach this question is to ask what 2 could
not be. The first thing is that, given the above proof and others like
it, we cannot automatically think that a nominalising strategy that
might look promising for the natural numbers or the rationals will
√
work for 2 . Thus it might be thought that we could regard number
as an abstraction for our purposes from aggregates of individuals, as
in, five sheep, three goats—that this is a social fact, like their worth in
Realism, irrationality, and spinor spaces 25

a marketplace. I don’t say that such a nominalising strategy has any

√
real plausibility merely that it will not work for 2 , for no aggregate
of individuals has that number.
Secondly, it might be thought that some geometrical magnitudes—
√
lengths, areas and volumes—might be this number 2 . But this
cannot be right either. The hypotenuse of a right-angled isosceles
(RAI) triangle is not intrinsically any number at all, rational or irra-
tional. Thus if we start with an RAI triangle with catheti of unit length
√
then the hypotenuse will have the length 2 . But if we had chosen
instead to make the hypotenuse of unit length then the catheti of the
triangle will each be √12 , which is irrational. The same can be said,
mutatis mutandis, for areas and volumes. Whereas it might be plausi-
ble to think of things as having natural units—one goat, one sheep,
one neutron, etc.—this cannot be carried across to geometrical mag-
nitudes. And if there are no natural units for geometrical magnitudes
then no other such magnitude is intrinsically irrational either.6 It is for
this reason that, by Euclid’s time, the phenomenon revealed by the
Pythagorean proof was sometimes referred to as incommensurability.
This is a pair-wise relation. The catheti and the hypotenuse of a RAI
triangle cannot both be whole numbers or ratios of whole numbers:
one must fail, but it is an arbitrary choice which one is made to fail.
√
The consequence is that 2 cannot be identified with geometrical
magnitudes in an absolute sense.
The third form of Nominalism is the one that I regard as initially
the most plausible, and the one that was outlined in the first section,
√
above,. The trouble is that this view will not work either for 2 . This
is because there is no numerical expression—I must emphasise ‘nu-
√
merical’ to forestall the irrelevant objection that ‘ 2 ’ is itself such an

6The Planck length might be thought to be a candidate for such a fundamental unit
but it is not clear whether at this level the continuity of the space is destroyed as well.
26 Adrian Heathcote

expression—for this or any other irrational number. In fact this seems

to be how the Pythagoreans themselves understood their discovery:
that they had discovered numbers that were unsayable. Evidence for
this can be found in Plato’s statement in The Republic: such numbers
(or magnitudes) were arrheton (unspeakable or unsayable).7
In fact as late as Euclid, Heath reminds us that the term that is
normally translated as ‘rational’ was rheta, meaning sayable, and
the obvious root of arrheton. By contrast the word in Euclid that we
translate as ‘irrational’ was aloga which can have as many meanings
as that very loaded word logos—but will certainly include beyond
words.8
In saying that irrational numbers are unsayable we do not of
course mean (and nor did the Greeks mean) that there is no form
of words which will describe such numbers, for the expression ‘the
square root of two’ is obviously such an expression. The point is that
there is no finite expression in numerals that will do so. As Leibniz
put it, in his Dialogue on Human Freedom and the Origin of Evil,
√
of 1695 (Leibniz, 1989), such magnitudes as 2 are not expressible
in numbres exact, and even God could not find such an expression.
If we allow infinite forms of expression then we can think of these
numbers as limits, for example by the approximation method known
as anthyphyrasis, which was known in Plato’s time. And this in itself
leads to a continued fraction representation of these numbers, as
discovered by Pietro Cataldi, Brouncker, Wallis, and Euler. But all
of these means of expression are essentially infinite: there is no finite
expression in numerals, or numbres exact: it is in this sense that they

7 Additional evidence is provided by the title of a lost work of Democritus, Of Un-

sayable Straight Lines and Solids, noted by Diogenes Laertius. This is the earliest
known written work on the Pythagorean discovery, since the Pythagoreans themselves,
famously, committed nothing to writing.
8 Euclid in Heath translation (Euclid, 1956).
Realism, irrationality, and spinor spaces 27

are unsayable. Every schoolchild learns at least one manifestation

√
of this profound fact: the decimal representation of 2 would be an
infinite, non-recurring string of numerals. Cutting it off after any finite
√ √
length will give a rational number that is not equal to 2 . So 2 is
something beyond what we can express in numerals. The habitual
confusion between numerals and numbers that has given nominalism
its longevity is simply not available in this case.
√
To say that 2 is unsayable in numerals must also be to wonder
whether it is a number at all. This is the important ontological issue
to which we have become numb, but which was still very much
a live issue in the 19th Century. It is a familiar point that ‘number’
for the Greeks meant natural numbers, though they also understood
ratios of these natural numbers. So it is possible that Plato could have
√
said, cautiously, that there was something that was 2 but remained
agnostic as to whether it was number in a new sense of the term, or
whether it was some other kind of entity whose square was a number!
And yet there was at least one good argument for thinking of these
unsayable entities as numbers in a new sense: the square root of 4
is a number, namely 2; so the square root of 2 surely ought to be
something of the same kind, despite being ‘unsayable’. They marked
their caution by distinguishing between geometria, as the study that
encompasses these entities, and arithmos. There is some evidence
in the later dialogues that Plato was prepared to take the step of
expanding the concept of number to including these new entities, at
Epinomis 990d, for example.9

9 That Plato came at some time in his adulthood to be imbued with Pythagorean
concerns is standard, and many date this transition to the post-Republic period. But
precise dating is more difficult. Philodemus dates it as early as Plato’s 27th year. He
then says: I [Philodemus] wrote it up. ‘It had been recognised, however’, he says, ‘that,
during that time, the mathematical sciences were also greatly advanced, because Plato
was supervising (them) and posing problems that the mathematicians investigated
28 Adrian Heathcote

If there are entities without numerical names then those entities

cannot be collapsed into such names—and the proof that there are an
uncountable number of real numbers means that not every real number
can receive a name of any kind. Thus even if we allow ourselves to
make use of countable ad hoc names—as we do with ‘𝜋 ’ or ‘e ’
√
—or disguised definite descriptions—as we do with ‘ 2 ’—we still
have a significant problem. For with the numerals expressing the
natural numbers there come algorithms for the common arithmetical
operations. But there is no such natural extension of these algorithms
for these ad hoc names. How would Plato (or any mathematician
√
before the 19th Century) go about adding 2 and 𝜋 ? Can we be sure
√ √ √
that 2 × 3 = 6? In fact it was Dedekind who noted, as late as
1858, that it had never been proven but only assumed that for real
numbers
√ √ √
𝑎. 𝑏 = 𝑎𝑏.

(And the issue was not trivial, as this equation fails for complex num-
bers (see Waterhouse, 2012). So a formalist or fictionalist conception
of Nominalism—in which mathematics is just the manipulation of
symbols according to set rules—has to confront the fact that here we
have entities for which there can be no systematic naming procedure.
Moreover this must have been evident even in Plato’s day, for there is
a complete absence of discussion of adding or multiplying arbitrary
incommensurables.

with zeal. In this way, accordingly, this was the first time that issues related to the
theory of ratios reached [the peak of their development], and the same holds for the
problems related [to definition], since Eudoxus and his followers introduced changes
to the old-fashioned approach [of Hippocrates]. Geometry [too] made great progress.
For there were produced both the method of analysis and the examination of the limits
(of a problem) and geometry in general was much [advanced]; furthermore, in [optics]
and mechanics [. . . ]’ Philodemus History of the Philosophers in Kalligas et al (2020).
Realism, irrationality, and spinor spaces 29

3. Taking roots

It seems that we have in this reconstruction a quite solid argument for

a form of mathematical realism.

a) There exist mathematical entities for which there is no plausible

nominalist construal.
b) These entities figure in the measurement of space and time
intervals and curvature, but also in particular physical problems,
including those that require the use of calculus. Moreover their
properties explain certain things that are impossible: namely
the Delian cube problem, squaring the circle, etc.

In a sense we have here an indispensability argument. But this ‘in-

dispensability’ is quite targeted in this case, for it is not simply an
indispensibility to modern science, but has a more general cast: an
indispensibility to nature herself. For if the nature of irrational num-
bers is able to explain the impossibility of carrying out particular acts,
how does a nominalist or a fictionalist strategy have anything that can
equally explain that impossibility? After all, neither are invoking the
existence of any items not already available to the realist. They are
arguing for less, and so have fewer resources. As far as I’m aware
there is no answer to this in the existent literature. The only nominalist
strategy of which I’m aware that might have something to say here is
that of Hartry Field in his (1980). Field helps himself to a particular
space-time manifold model to argue that real numbers are unneces-
sary, but his argument is restricted just to explaining positive metrical
facts, not all facts. I think his argument fails in general (I take it up in
section 5) and if it fails there is nothing to replace it.
30 Adrian Heathcote

And yet though this gives us a realism of the real numbers—it

does not in itself provide us with a reason to be realist about other
mathematical entities.
But the way we go beyond this beginning point is exactly the same
as the way mathematics itself evolved beyond this beginning point.
Euclid is the germ from which mathematics grew, by demonstration
from axioms which are self-evident. For 1700 years mathematics con-
sisted of furthering the work of Euclid by enlarging on the subjects of
geometry, arithmetic and analysis. Abstraction led to algebra, whether
in whole number solutions, as in that of Diophantus, or generally
in real numbers. But whether mathematics was furthered by solving
equations or giving proofs, the method by which mathematical knowl-
edge was gained was hardly a mystical intuition. Mathematical truths
are known by proof and calculation.
The stability of mathematical ontology up to the 15th Century
and the revival of Platonism and the re-establishment of the Academy
in Florence under Marsilio Ficino and Cosimo de Medici, laid the
ground for the next expansion: the discovery of the complex numbers.
The tale has been told often enough of the discovery of the method
of solving cubic equations by Tartaglia and its theft and publishing by
Cardano in 1545. The interpretation of the root of −1 as a geometric
mean of 1 and −1 obtained by solving
1 𝑥
=
𝑥 −1
and the interpretation of this, geometrically, as a mean proportional
perpendicular to the ordinary number line gives us ‘two-dimensional
numbers’, removing linearity as an essential condition of what it is
to be a number. Again, the mathematical aspect of the discovery of
complex numbers has been well-described elsewhere, but what of the
philosophical significance?
Realism, irrationality, and spinor spaces 31

The striking thing about the way complex numbers arise in the
solution to the cubic is that they seem to force themselves upon us.
We are looking for real solutions to a cubic equation, which itself has
only real coefficients, and yet complex numbers arise naturally on the
way to the real number solutions. Thus consider this example, from
Bombelli’s L’ Algebra: x3 = 15 x + 4. The three roots of this equation
√ √
are 4, −2 − 3, −2 + 3. They can be found by solving this equation
from Scipione Dal Ferro, with b = 15, and c = 4:
√︃ √︂ √︃ √︂
3 𝑐 𝑐2 𝑏3 3 𝑐 𝑐2 𝑏3
𝑥= + − + − − .
2 4 27 2 4 27

This will give us, on substitution:

√︀
3 √ √︀
3 √
𝑥= 2 + −121 + 2 − −121

or
√
3
√
3
𝑥= 2 + 11𝑖 + 2 − 11𝑖

where each cube root has three solutions. One of these, 2 + i along
with its conjugate 2 − i, Bombelli must have found, since it yields the
root 4, which he gives as a solution to the equation. (Bombelli would
have been inclined to discard the negative roots.)
The philosophical puzzle that Bombelli faced was this: the roots
of the equation are acceptable numbers, or at the very least, one of
them is; but the method by which we reach them involves taking
the cube roots of numbers that appear unreal or “sophistical”. And
the cube roots themselves are also unreal or sophistical. But it is
only by adding together these unreal numbers (in conjugate pairs)
that we reach the roots, that we must take seriously. For Bombelli
the puzzle must have verged upon paradox: for he did not regard
negative numbers as proper—by contrast he had no problem with
32 Adrian Heathcote

irrational numbers—and yet he was taking the square root of negative

numbers, and then taking the cube root of the complex radicals that
resulted—and then adding them pairwise.10 He declared this discovery
as the discovery of a new kind of cubic radical and said that he had
a geometrical proof of it. He says:

This kind of root has in its calculation different operations than

the others and has a different name. . . [It] will seem to most
people more sophistic than real. This was the opinion I held
too until I found its geometrical proof (translated in Federica
La Nave and Barry Mazur’s (2002)).

This geometric proof of Dal Ferro’s equation appears late in

Bombelli’s work and resembles the geometric proofs of the existence
of irrationals: in a sense complex numbers stand to irrationals as Dal
Ferro’s equation stands to Pythagoras’s Theorem—they both emerge
as surprising solutions given well-recognised inputs.11 However it was
not for another 100 years, when Wallis and then De Moivre showed
√
that −1 could be not just be proven to exist but also given a repre-
sentation in the Euclidean plane, the mis-named Argand plane, that its
acceptance was assured. But they—i.e. complex numbers—come to
us as a natural extension of our previous ontological commitments—
they were not ‘posited’ for the purposes of doing physics, or whatever,

10
√ √
His notation for −1 was R (0 · m · 1) which translates directly to 0 − 1 with
‘m’ standing for ‘minus’— thus neatly avoiding making the negative sign an adjectival
modifier. Note also that there are nine pairs that could be summed, and it requires some
clarity to realise that only three of those pairs, the conjugates, are relevant for finding
the roots.
11 The geometric proof is broken down a little in La Nave and Mazur (2002, 17ff). See

also Barry Mazur’s (2004), which tells the story of Bombelli’s imaginative leap.
Realism, irrationality, and spinor spaces 33

they were instead a discovery that emerged naturally from pursuing

ordinary mathematics. And it is this that gives one the confidence that
they exist.12
En passant this helps to solve another puzzle. It has sometimes
been said that the discovery that our physical space is not Euclidean
but instead has a Riemannian curvature shows that Euclidean ge-
ometry is “wrong”. This, I think, is a mis-saying. The geometrical
representation of the complex numbers shows that the axioms for two-
dimensional Euclidean geometry are instantiated after all. They are
just not instantiated in the way one might have thought. And once we
have an instantiation for Euclidean space then we get linear algebra
and operators all as part of the machinery for the description of that
space. The rich connections between Euclidean geometry and the real
and complex numbers have been thoroughly explored, and need no
further comment. Again, this is an issue we come back to.13
Our realism, or platonism, has taken us as far as complex numbers
and linear geometry with no reliance on the usefulness of mathematics
to physics—and Bombelli died 60 years before the appearance of even
Galileo’s Dialogue Concerning the Two Chief World Systems. Most
curiously, the expansion of the mathematical ontology—or, to put it
more accurately, the realisation that there was more ontology implicit
in the initial commitment to whole numbers than had been realised—
in both Pythagoras and Cardano-Tartaglia-Bombelli—involved taking
roots. Once again: taking roots has been ontologically ampliative. In

12 Thus I am here resisting the idea that the indispensibility of mathematics be given
a pragmatic cast, as though it were a tool of an engineer with an Aristotelian bent (q.v.,
Newstead and Franklin, 2012).
13 See for example Liang Shin-Hahn (1994); also Kaplansky (2003) (a reedition of

1969).
34 Adrian Heathcote

fact had the ancients been prepared from the outset to countenance
negative numbers then the process of taking roots might have led
directly to the complex numbers two millennia earlier.
Complex numbers are used routinely in quantum mechanics—but
do we have any evidence that their use is unavoidable? Until recently
the answer would have seemed to be ‘no’, for it always looked pos-
sible to translate standard quantum mechanics on the complex field
(CQM) into a more cumbersome real number form (which we will
abbreviate to RQM). This is hardly any form of nominalism, but it has
been a standard suggestion made against being realists about complex
numbers. This situation may have changed recently by a paper that
argues that there are situations in CQM that cannot be explained in
RQM (Renou et al., 2021). The gist of the argument is that if we take
three individuals, Alice, Bob and Charlie, and have two entangled
photons shared between Alice and Bob, and another two shared be-
tween Bob and Charlie: when Bob measures the two particles he has
received the entanglement is transferred to one between Alice and
Charlie, even though they have not received particles from a common
source. The claim of Renou et al. (Renou et al., 2021) is that this
transfer of entanglement can’t be explained in RQM, though it can
be explained in CQM. They calculate an entanglement coefficient,
√
based on the Clauser-Horne-Shimony-Holt inequality, of 6 2 , which
is higher than the maximum attainable by RQM. There is also an
experimental protocol that could test this difference. If the test were
to come out as the authors believe then complex numbers would not
after all be eliminable in favour of real numbers.
If this is so, what we have is a mathematical discovery that is
essential for physics being made well before that physics came into
Realism, irrationality, and spinor spaces 35

existence. It would be hard in this circumstance not to come to the

conclusion that mathematical discoveries are of something real that
are laying the groundwork for us to make such physical discoveries.
A very similar case is provided by the quaternions. Hamilton’s
construction of these was designed to be by analogy with the complex
numbers: he wished to find a four-dimensional analogue of them to
represent spatial rotation. But it was not forced by the solution of any
existing equations or problems in mainstream physics or mathematics.
So we once had no reason to believe that they exist—only that they
could possibly exist. Nevertheless, subsequently, we may feel quite
differently: W. K. Clifford’s use of them in what we now call Clifford
algebra, and the role that they play in the theory of spinors, may
convince us that Hamilton’s instincts were right, against the critics of
the day. This is the issue we take up in the next section.14

4. Spinors

We can find an even more significant discovery that affords a bet-

ter example of mathematics preceding the physics for which it is
indispensible.
In his (1913) Élie Cartan discovered an entirely new representa-
tion of the orthogonal Lie Algebra SO(3) which could not be obtained
from vector representations. This was, again, a discovery in pure
mathematics—following on from previous discoveries in transfor-
mation groups: there were entities which transformed in a wholly
unexpected way. Quite separately, however, Wolfgang Pauli began
to employ these entities in quantum mechanics in 1927 as a way of
describing electron spin (followed, independently, by Dirac for the

14 For the fraught history of quaternions see Simon Altmann’s (1989).

36 Adrian Heathcote

relativistic electron in 1928) and the mathematical entities were then

named after their physical manifestations: spinors. R. Brauer and H.
Weyl described the mathematical theory of these entities in a paper in
1935, without knowledge it seems of Clifford algebra, and then Cartan
followed with a fuller monograph in 1937—making full reference
to Grassmann’s exterior algebra and Clifford’s usage of it. In Weyl’s
Classical Groups (1939), the fuller picture is given also.15 Thus we
have from Weyl (1939) the derivation of the spin representation. ‘In-
stead of the projective we have thus obtained an ordinary though
double-valued representation ±𝑆 (𝑜) of degree 2𝜈 , called the spin
representation.’ Significantly, he goes on:
The normalization requires the possibility of extracting
square roots. The constructions in Euclidean geometry with
ruler and compass are algebraically equivalent to the four
species and the extraction of square roots. A field in which
every quadratic equation 𝑥2 − 𝜌 = 0 is solvable may therefore
be called a Euclidean field. Our result is then that in every
Euclidean field we can construct the spin representation; the
Euclidean nature of the field is essential. The orthogonal trans-
formations are the automorphisms of Euclidean vector space.
Only with the spinors do we strike that level in the theory of its
representations on which Euclid himself, flourishing ruler and
compass, so deftly moves in the realm of geometric figures. In
some way Euclid’s geometry must be deeply connected with
the existence of the spin representation (Weyl, 1939, p.273).

What might Weyl have meant by this enigmatic final remark? We

find it echoed by Michael Atiyah. ‘No one fully understands spinors.

15 Brauer and Weyl (1935, pp.425–449). For the English translation of Cartan’s mono-
graph: (Cartan, 1966). B.L. van der Waerden was the important link between Ehren-
fest’s physics group and the mathematical community in the early 1930’s: it was the
latter who simplified and made accessible the mathematics. See Veblen (1933) and
(1934), also Payne (1952).
Realism, irrationality, and spinor spaces 37

Their algebra is formally understood but their general significance is

mysterious. In some sense they describe the ‘square root’ of geometry
and, just as understanding the square root of −1 took centuries, the
same might be true of spinors.’ (quoted in Farmelo (2009)).16 What is
the ‘square root of geometry’?
Isotropic vectors are those whose ‘length’—as given by the square
of the modulus—is zero. So let x = (𝑥1 , 𝑥2 , 𝑥3 ) be an isotropic vector
in a three-dimensional space. In fact we will specify that the space
is C3 to make the connection with the physics more apparent—thus
each of the components is a complex number. The isotropic vectors
form a two-dimensional surface in C3 , and for each we will have

𝑥21 + 𝑥22 + 𝑥23 = 0.

Each such isotropic vector has associated with it two numbers 𝜉0 and
𝜉1 given as solutions to the following three equations:

𝑥1 = 𝜉02 − 𝜉12 ,

𝑥2 = 𝑖(𝜉02 + 𝜉12 ),

𝑥3 = −2𝜉0 𝜉1 ,
where these are of the form
√︂ √︂
𝑥1 − 𝑖𝑥2 −𝑥1 − 𝑖𝑥2
𝜉0 = ± and 𝜉1 = ± .
2 2
These two numbers parameterize the two-dimensional surface of
isotropic vectors. The vector
⎛ ⎞
𝜉0
⎝ ⎠
𝜉1

16In a direct reference, Atiyah, in his 2013 conference lecture “What is a Spinor?”
quoted Weyl’s line verbatim.
38 Adrian Heathcote

is a spinor. But as with Bombelli’s solution to Dal Ferro’s formula

there are two choices, depending on the sign, as the solutions come in
yoked pairs (again the cross terms are discarded). So we also have
⎛ ⎞
−𝜉0
⎝ ⎠
−𝜉1
√
as a second solution, analogous to the partnering of −1 and
√
− −1 .
Though Atiyah spoke of spinors as being ‘square roots’ of
(isotropic) vectors, Cartan himself refers to them as “polarisations”—
“en quelque sorte un vecteur isotrope orienté ou polarisé”, where
a rotation of this vector through 2𝜋 changes this polarisation of the
isotropic vector (Cartan, 1966, p.42). They are of course now ubiq-
uitous in physics since fermion states are spinors. These are not
unknown in relativity theory either—the light cone is represented by
isotropic vectors and has associated with it spinors (with real compo-
nents) which are time-like. This was the point of view emphasised by
Cartan in his 1937 lectures, with particular emphasis on Minkowskian
geometry. Since Brauer and Weyl in (1935) had given an algebraic
view, Cartan wanted to emphasise their relation to space-time geome-
try. Thus he presented
[. . . ] a purely geometrical definition of these mathematical
entities: because of this geometrical origin, the matrices used
by physicists in Quantum Mechanics appear of their own
accord, and we can grasp the profound origin of the property,
possessed by Clifford algebras, of representing rotations in
space having any number of dimensions. (Cartan (1966) from
his Introduction.)

But, with respect to his conception of spinors, he also pointed to the

impossibility of using the usual coordinate transformation techniques
Realism, irrationality, and spinor spaces 39

in Riemannian geometry (a remark that was sometimes mistakenly

construed as an impossibility proof of introducing spinors into general
relativity).
Spinors are closely related to the Atiyah-Singer Index theorem
and K-Theory, the Seiberg-Witten theory and Alain Connes’ non-
commutative geometry. Roger Penrose has made them the centrepiece
of his proposed unification scheme for relativity and quantum mechan-
ics in Twistor theory (Penrose, 2004). Their fundamental character
would be hard to overestimate—and yet they emerged, firstly, from
pure mathematics, only to (independently) come, some 13 years later,
to represent a property that had no macroscopic visualisation: a hith-
erto unsuspected property of matter that arose first from the abstract
study of Lie groups—from the Lie group SO(3) and its double cover
SU(2). This is surely one of the most dramatic and least heralded
examples of the uncovering of mathematical structures in nature. And
here the mathematics seems very close to being directly physically
detectable in the form of spin eigenvalues. And due to the character
of the double cover SU(2) spinors have the remarkable property that
if we pick an isotropic vector and rotate it through 2𝜋 it returns to its
original position but the spinor is only rotated through 𝜋 and its sign is
reversed. It takes a rotation of 4𝜋 to bring it back to its original state. It
is argued in Christian (2014) that this is also measurable.17 Moreover
it is remarkable that the spin values of the fermions and bosons arise
directly from the dimension of the irreducible representations of the
Lie algebra sl(3), which is the Lie algebra of the groups SO(3) and
SU(2)—the former giving the spin values for bosons and the latter for
fermions.

17See Penrose and Rindler (1987; 1988). Also Claude Chevalley (1997), particularly
the afterword by J.-P. Bourguignon; also Lounesto (2001).
40 Adrian Heathcote

The non-classical nature of a spinor’s double-rotational invariance

is surprising and constitutes a challenge to the idea that particles can
be seen as physical objects in the classical manner. Despite this,
and acknowledging that it represents only a partial solution to the
geometrical problem, Penrose has ingeniously utilised the properties
of the Riemann Sphere P(H 2 ) to give a graphical representation
of the pure states of spin. It is when one moves to higher fermionic
spin states that this picture—the Majorana picture—becomes highly
non-classical and defies ready visualisation. Penrose pointed out that
as we aggregate matter to form higher spin values that there is no
convergence to the classical picture, rather the opposite.
[. . . ] we see that a randomly chosen quantum system with
a large angular momentum (large j value) has a state defined
by a Majorana description consisting of 2j points more-or-less
randomly peppered about the sphere 𝑆 2 . This bears no resem-
blance to the classical angular momentum state of a system of
large angular momentum, despite the common impression that
a quantum system with large values for its quantum numbers
should approximate a classical system! [. . . ] The answer is
that almost all ‘large’ quantum states do not resemble classi-
cal ones (Penrose, 2004, p.566; also see Penrose and Rindler,
1987; 1988).

But despite defying ready geometric visualisation, spinors are

required in quantum theory. Since the work of Cartan, Weyl, and then
Chevalley in the 1950’s it has become clear that the natural home for
a discussion of spinors is Clifford Algebra. And within the Clifford
Algebra in which the simplest expression of quantum mechanical
spin is representable, the 8-dimensional algebra usually denoted Cl 3 ,
the real numbers and the complex numbers are naturally represented
as sub-algebras. Thus, spinors represent a culmination of algebraic
structure within the structures applicable in physics, that includes
Realism, irrationality, and spinor spaces 41

the real and complex numbers, and also the quaternions. And it is
the unit quaternions that are the spinors as defined by Pauli. Thus
Clifford Algebras encapsulate and relate together these seemingly
different mathematical structures—all of which are intimately related
to our most successful physical theories and in the case of the real and
complex numbers, spinors, and quaternions, actually preceded them.
We can close the circle on the progression that we have been
noting here: from right angled triangles to the Pythagorean under-
standing of irrationality and the real numbers, to complex numbers,
to spinors, by mentioning a remarkable fact: Pythagorean triples can
be understood as generating spinors defined on the null vectors of
Z3 . This is due to the mapping induced by the Euclidean parameters
(𝑝, 𝑞), with 𝑝 > 𝑞 , to the Pythagorean triples (𝑥, 𝑦, 𝑧) by

𝑥 = 𝑝2 − 𝑞 2 , 𝑦 = 2𝑝𝑞, 𝑧 = 𝑝2 + 𝑞 2 .

At least one of the numbers (𝑥, 𝑦) must be even. The primitive

Pythagorean triples are those that are mutually prime. A standard
Pythagorean triple is one which is either primitive with 𝑧 positive and
𝑦 even, or ( 𝑥2 , 𝑦2 , 𝑧2 ) is primitive and 𝑦2 is odd. Thus the triple (3, 4, 5)
is standard, whereas (4, 3, 5) is not. Then, it is provable that for every
standard Pythagorean triple there is a pair of Euclidean parameters
that are relatively prime which generate the Pythagorean triple. This is
then a one-to-one correspondence (bijection) between the directions
in Z2 and the null directions in Z3 .18
Euclid’s discovery of the parameterisation of Pythagorean triples
may be viewed then as the first recorded use of a spinor space.

18See Trautman (1998) Proposition 1. These ideas are developed in greater detail in
Kocik (2007), though without acknowledging Trautman’s prior work. Kocik links this
with quasi-quaternions and the Apollonian Gasket.
42 Adrian Heathcote

This in turn is related to complex numbers: 𝑐 = 𝑝 + 𝑞𝑖, since the

norm is equal to 𝑐𝑐*, the complex number multiplied by its conjugate,
which is
𝑝2 + 𝑞 2 .

And the square of the complex number is

(𝑝2 − 𝑞 2 ) + 2𝑝𝑞𝑖.

Thus the squares of certain integer complex numbers generate

Pythagorean triples. Or, to put it another way, Pythagorean triples have
square roots that are integer complex numbers. A comparison with
the immediately preceding discussion of isotropic vectors shows that
Euclid’s three equations for Pythagorean triples are analogous to the
equations that define a spinor in Cartan’s formulation. Pythagorean
triples are spinors in Z2 ! As Kocik (2007) puts it: ‘Euclid’s discovery
of the parameterisation of Pythagorean triples may be viewed then as
the first recorded use of a spinor space.’
This appears to vindicate Weyl’s mysterious remark.19 But it also
emphasises that there is a connection between the metric on the space
and the definition of spinors—so that the latter actually requires the
former. This dependence is further discussed in Bär et al. (2005) and
Bourguinon et al (2015).
Let us return briefly to Penrose’s idea of the centrality of the
Riemann sphere. As noted, he pointed out that a spin-½ particle can
have the possible directions in which its spin can be measured mapped
to the Riemann sphere. But he then said:

19 Of course we might also add, for further evidence, that the square root of the classical
Laplacian is the Dirac operator of relativistic quantum mechanics—and this takes us
to the Lorentz invariant spinors of Cartan. There is thus a sense, not entirely figurative,
in which quantum mechanics is the square root of classical mechanics, as suggested
by Penrose.
Realism, irrationality, and spinor spaces 43

Although quantum amplitudes seem to be very abstract

things, having this strange ‘square root’ relation to a proba-
bility, they actually have close associations with space-time
geometry (Penrose, 2000, p.230).

To make this connection he noted that being situated at a point in

space the light cone at that point can also be represented by a Riemann
sphere. This sphere represents all of the light like rays that pass
through the observer’s point in space. This Riemann sphere is then
conformally deformed if we pass to another observer passing through
that same point with a different velocity, Thus the non-reflective
Lorentz transformations can be represented by complex conformal
transformations of the Riemann sphere. It would be interesting to
consider that these different usages of the Riemann sphere could be
unified by Cartan’s geometrical picture of spinors as square roots of
null vectors.20

5. Realism defended

The enlargement of mathematical ontology from Pythagoras through

to Cartan and Weyl is properly the uncovering of structure al-
ready present, and uncovered through the process of doing ordinary
mathematics—solving equations, constructing proofs, analysing exist-
ing mathematical structures. And through this process mathematicians
have given us an understanding of real numbers and analysis, includ-
ing differential geometry; complex numbers and their associated struc-
tures in geometry and algebra; and spinors and their structures. In
these three cases the mathematical structures preceded, sometimes by
centuries, their application in physics.

20 Penrose (2004) does not reference Cartan in this context.

44 Adrian Heathcote

We can thus see the danger in an over-reliance on the indispens-

ability thesis. There is a strongly pragmatist construal of this thesis
that would have it that the only reason we should believe in math-
ematical entities is their usefulness in physical explanation—with
the implication that if they had not found an application in physi-
cal explanation we would not have reason to believe in them. This
does an injustice to the very thing that makes mathematical episte-
mology unique: proof. A far more compelling fact about the use of
mathematics in physics is that the mathematical discoveries made
by entirely different methods often precede the discovery that they
can be found also in the natural world. It is this that should keep the
Nominalist awake at night. But we should accept a more modest role
for indispensability: that physics is capable of providing a layer of
additional confirmation that mathematical structures and entities exist,
and moreover that this existence should not be regarded as an abstract
matter, for they are part of the fabric of the Universe.
Thus let us consider the most well-developed nominalist view:
that proposed by Hartry Field in his (1980). The central idea is to
take congruences on a Newtonian space as giving one all the ‘num-
bers’ that we need. And yet I think it misses the mark. As suggested
earlier, if the nominalist is permitted to help himself to space-time
as a flat 4-dimensional differentiable manifold with a metric struc-
ture then he has thereby helped himself to the real numbers already,
both in the metric and also in the differentiable structure. For an n-
dimensional differentiable manifold is locally isomorphic to R𝑛 .21
In fact a 4-dimensional, not necessarily flat, differentiable manifold
proves to be unlucky for Field and nominalists generally as it is the
only dimension for which there can exist an infinite number of dis-

21This style of criticism of Field was signalled early on by Michael D. Resnik in his
review of Field’s book: (Resnik, 1983; also Resnik, 1985; see also Steiner, 1998).
Realism, irrationality, and spinor spaces 45

tinct quasi-conformal structures—and thus there can be more than

one way to determine the local mappings to R4 that are conformally
inequivalent (Donaldson and Sullivan, 1989). These are not simply
many different metrics definable on a differentiable manifold—which
would be a trivial point and would not distinguish 4-manifolds. Rather
the quasi-conformal structures are distinct in that no finite amount of
stretching or shrinking of the metric will deform one quasi-conformal
manifold into another, despite being topologically identical. (Guided
as we are by 2- and 3-dimensional topology this seems impossible
to visualise.) The problem for Field is that these infinite possibili-
ties are precisely the kind of abstracta that his nominalism cannot
countenance.
If the space-time is Newtonian (as it is for Field) then the metric is
globally singular though well-defined on the time-like fibrations—this
alone creates complications since then his congruence relations are
only defined on the fibrations. If it is Minkowskian then it is globally
well-defined everywhere.22 Field is of necessity a substantivalist about
space-time geometry but I cannot see how comparatives will allow
him to give a Nominalist construal of the light cone structure, since all
points that are light-like separated have 0 distance from one another
by the Lorentz-signature metric, even when they are collinear! This by
itself refutes the idea that congruence can be a nominalist substitute
for the role that the metric structure plays! Since Minkowski space-
time is a more realistic space-time structure than Field’s preferred
euclidean space this seems definitive.
But I’d like to sketch an ancilliary argument of a different kind,
which suggests that Field’s strategy does not do away with numbers
in the way he suggests, even on Euclidean space. Suppose that there

22
This leads David Malament in his review (1982) to shift Field’s case to a Klein-
Gordon scalar field theory.
46 Adrian Heathcote

were two four-dimensional manifolds, one with its metrical structure

determined by a mapping to R4 and the other with the a mapping
to the quaternions H or even to C × C. According to Field’s nom-
inalism these spaces are acceptable because they can be construed
substantivally, though the metrical structures are not taken to be sub-
stantive, because they involve numbers. So his plan is to eliminate
these metrical structures using his reformulation thesis in favour of
segment-congruences. But this presents him with a dilemma: either
the results of this elimination gives us the same ‘space-time’, or they
are different. If they are the same then the reformulation has eliminated
crucial information—because multiplication (and therefore segment
length) acts differently in these cases—but if they are different then
the metrical structures are still present in an implicit form: we have
simply stopped using useful numerical words! We could run this same
argument with a comparison of R4 with Minkowski space-time, or
with a different signature metric entirely, such as (+ + - -), or, most
significantly, with a Kähler 4-manifold in which there is more than
one metric-like structure.
If real numbers are smuggled in in the form of geometric structure
then the nominalist, though helping himself to a lot, has still not got
enough even for the simplest cases of quantum mechanics. If we
consider the Hilbert space as a space of the possible states of a system
then it is clear that even in the simplest case of C2 —for a spin-half
spinor space—it is not reducible to anything that Field is prepared
to countenance—for despite being topologically identical to R4 —
which Field needs for the purpose of his space-time structure—it
is precisely unlike R4 in its metrical features. And for the Hilbert
space of the spin-1 particle there is simply nothing available at all.
The problems are then only compounded from this point on. Once
we begin to consider quantum field theory we must consider spaces
Realism, irrationality, and spinor spaces 47

of operators that are defined on each of the space-time points. So

let us consider the noncommuting operators of the electro-magnetic
field: then the algebra will be an infinite-dimensional noncommutative
algebra. Dispensibilist Instrumentalism has no hope in this case, nor
has it ever been attempted.23
The Indispensibilist Instrumentalist might accept all of this as
evidence of the indispensability of said mathematics but insist that
we can think of the mathematics as merely “indexing” the physical
facts. The term ‘indexing’ comes from Melia (2000) and is meant
to cover the use of real numbers for distances as well as other cases
of measured magnitudes. However it is not at all clear what else it
is meant to cover and without a very clear recipe for applying the
term the charge of question begging will be hard to avoid (see Daly
and Langford (2009) for a defence of this way of understanding
Indispensibilist Instrumentalism).24
Thus it is hard to see how we can account for dimensionless
physical constants—such as Summerfeld’s fine structure constant

23 Of course, we can accept, with Field, that there is no canonical natural isomorphic
mapping of an n-manifold to R4 . But that is less than Field needs, for we can allow
that the metric is defined only up to a scale factor without abandoning the idea that
the metric is a part of the space. The metric simply becomes an equivalence class
of numerical assignments, equivalent up to a scalar factor. But by only allowing
congruence classes of intervals Field ends up with less than this—and here we come
back to one of the main themes of this paper—for he cannot capture the facts about
incommensurable magnitudes that so impressed the Greeks. Thus consider again the
√
1:1: 2 triangle. Congruence classes will allow Field to say that the two catheti are the
same length, but not that no integer that is assigned to that class will allow an integer to
be assigned to the hypotenuse, or vice versa. Incommensurability is a pairwise, metrical
relation, and it is entirely intrinsic to the space. So it holds for every coordinatization
in the equivalence class that defines the metric.
24 I say nothing in this paper about structuralism, as I’ve discussed it elsewhere—

see Heathcote (2014). In its anti-realist form structuralism is unable to address the
objections made here.
48 Adrian Heathcote

𝛼 = 0.0072973525693 . . ., first introduced in 1917. The constant has

the (or one) meaning:

1 𝑒2
𝛼 = .
4𝜋𝑒0 ℏ𝑐

Here 𝑒0 is the electric constant and 𝑒2 is the square of the elementary

charge of an electron. The value of the constant has been measured
very accurately—and the accuracy is always improving, but it does
not seem to be related to any known mathematical constants, and note
that it isn’t clear, and may never be clear, whether it is rational or
irrational.25 And yet, it has been argued that if 𝛼 were different by
even a small amount then the Universe would not exist: matter as we
know it would not exist. However it is not its precise value that is our
concern here, but simply the fact that it is a dimensionless number.
For the nominalist view is that numbers don’t exist, and thus that 𝛼
does not exist either. But if that is the case then, never mind its exact
value, no such value exists—and so matter can’t exist. No form of
nominalism of which I’m aware has made an attempt to deal with
this problem of dimensionless constants such as 𝛼—and no strategy
suggests itself. That is the realist argument in its starkest form, and
indeed may summarise the point of this paper: either numbers exist
or nothing exists.26

25 The fine structure constant is often given in the reciprocal form 𝛼−1 =

137.035999206. . .
26 As Wolfgang Pauli is alleged to have said: ‘When I die my first question to the Devil

will be: What is the meaning of the fine structure constant?’ Of course there are other
dimensionless physical constants besides 𝛼 that could make the same point.
Realism, irrationality, and spinor spaces 49

But, as hinted at earlier, I believe we can find a simpler case, with

ancient and venerable Platonic credentials, that seems rather clearly
to not be a case of mere indexing. And it is one that is equally as hard
for any form of nominalism that is currently espoused.27
The argument is as follows. Premise 1: Whether an action can be
performed, or a task completed, has a determinate truth value: one
either can or can’t. Premise 2: Whatever facts the ability to perform the
act or complete the task depends upon must likewise be determinate.
But consider the task set by the Delphic oracle to the Delians: they
were required to double the size of their altar—which was cubic
shaped. And let us suppose, as Plato apparently did, when the Delians
approached him on the matter, that this doubling of the cube must be
done only within constructive geometry, that is with straightedge and
compass, anything else being merely approximate.28 The doubling of
√
the cube requires finding 3 2 which is irrational (the proof is an easy
√
generalisation of some of the proofs of the irrationality of 2 ). But it
is also a non-constructible number—as proven by Wantzel in 1837.
And this means that there is no way to perform the action required
√
by the oracle. So it is false that 3 2 is constructible and so false that
the Delians task can be performed. The same argument can be run
using squaring the circle as the example, where the impossibility

27 Jody Azzouni has recently resurrected, in his (2006) and (2010), a form of pure
fictionalism about mathematics—mixed with a form of social constructionism—that
seems particularly vulnerable to this challenge, as it makes no attempt to deal with
the mathematics that occurs in physics and is content to discuss the counting and
computation of natural numbers(see Batterman, 2010).
28 Thus note Plutarch’s comment on this: ‘And therefore Plato himself dislikes Eudoxus,

Archytas, and Menaechmus for endeavouring to bring down the doubling of the cube
to mechanical operations; for by this means all that was good in geometry would be
lost and corrupted, it falling back again to sensible things, and not rising upward and
considering immaterial and immortal images [. . . ] .’ Platonic Questions, Quest. 2,
Moralia.
50 Adrian Heathcote

depends upon the transcendental character of 𝜋 , which implies that

it too is non-constructible.29 The point is that mathematics is not
just, as a discipline, indispensable to science, it is that mathematical
facts constrain and determine physical facts, and cannot easily be
distinguished from them. Thus, as another example, it is the topology
of the 2-sphere that determines that there must be some point on
Earth where the wind does not blow. It is impossible to partition
explanation into the physical versus the mathematical in a way that
leaves Nominalism with any clear content. Once we have let in what
is needed for physical explanation then mathematics has been let in as
well. This is particularly the case with the structures chosen here: the
division algebras and the spinor structures. Mathematics and physics
seem to have converged.

6. The royal road to ontology

It is time to take stock.

In the process of taking roots we have jumped from a discrete
structure to continuous structures, in other words to geometry. In the
first instance this led us to the real numbers, via incommensurable
magnitudes and irrational numbers. Then in a second step we were led
to the complex numbers and their richer geometry. And then, through
complex numbers, Clifford algebras, and quaternions, we arrived at
spinors. I’ve argued that there is no plausible nominalist strategy that
can account for these structures: Field’s nominalist strategy won’t
work and—even if its problems could be set aside (as I believe they

29 As proven by Lindemann in 1882. The impossibility of squaring the circle was

probably known by the time Plato was writing: Aristophanes ridicules circle-squarers
in The Birds.
Realism, irrationality, and spinor spaces 51

cannot)—we would confront the problem of the dimensionless con-

stants. This latter problem defeats even a putative structuralist solution.
Nor is a narrow indispensibilist explanation plausible. My suggestion
in this paper has been that the major steps of this progress would
warrant a realistic attitude to these entities even if one could lay aside
the application of this mathematics to physics.
So the process of taking roots turns out to be ontologically amplia-
tive, and resists nominalistic reduction. Should we find this surprising?
One might suggest, aphoristically, that platonism manifests itself in
its most irresistible form as geometry. In support we may quote Shing-
Tung Yau, the inventor of Calabi-Yau manifolds, on the importance
of geometry:

Since the time of the Greek mathematicians, geometry

has always been at the centre of science. Scientists cannot
resist explaining natural phenomena in terms of the language
of geometry. Indeed, it is reasonable to consider geometric
objects as part of nature. Practically all elegant theorems in ge-
ometry have found applications in classical or modern physics
(Shing-Tung, 2000, p.253).

This of course is not to seek to take anything away from algebra, or

to suggest that arguments for realism do not extend to algebras. How
could they not when there is such a close relationship between algebra
and geometry? If geometry may be likened to the face, then algebra is
the mind behind the face. As Kähler said (in a philosophical essay):
“[. . . ] one must interpret the development of algebra as the revelation
of the realm of ideas postulated by Plato” (Kähler, 2003).
Thus this argument for mathematical realism gives precedence
to the reals over the integers, and to the complex numbers over the
reals. This is not to say that nominalism can easily deal with the
integers—I believe that even here it must fail. But in mathematical
52 Adrian Heathcote

terms the integers are now just one example of a commutative ring,
one among an infinite number of others—and quantum mechanics
has directed our attention to the non-commutative rings as possibly
equally or more fundamental. The primacy of the three associative di-
vision algebras in mathematical explanation — the reals, the complex
numbers and the quaternions — is what I mean by saying that these
‘almost geometrical structures’ are the primary basis for mathematical
realism, a meaning that is in accord with Plato’s own emphasis on the
importance of geometry. These division algebras and their associated
higher structures, such as Clifford Algebras, or spin representations,
are structures about which we must be realist.30 It is here that the
evidence is most irresistible. Indeed, if we turn the matter around, we
could say this: the only plausible explanation for physics continually
using the seemingly abstract mathematical structures uncovered by
mathematics is that our universe contains those mathematical facts
as generalised, non-local, parts of itself. In short: as ‘geometry’. My
historical conjecture is that this was itself Plato’s original insight,
inscribed on the entrance to the Academy: Let No-one Unskilled in
Geometry Enter Here.

Acknowledgement: I wish to express my warm thanks to the read-

ers for the journal who offered useful suggestions.

30 In this context, the importance of group representation theory in quantum physics is

worth emphasising. For here we take an often complicated non-linear algebraic object
like a group and we consider it under the aspect of a geometric object by considering
a homomorphism to a vector space. That this is especially fruitful has been argued
often, as far back as Weyl (1931) or Wigner (1939). For additional comments on this
see Heathcote (2021).
Realism, irrationality, and spinor spaces 53

Declaration: The author declares that there are no conflicts of

interest, no funding issues, and no ethics issues involved with this
paper.

Bibliography

Altmann, S.L., 1989. Hamilton, Rodrigues, and the quaternion scandal.

Mathematics Magazine, 62(5), pp.291–308. https://doi.org/10.1080/
0025570X.1989.11977459.
Azzouni, J., 2006. Deflating Existential Consequence: A Case for Nominalism.
Oxford University Press.
Azzouni, J., 2010. Talking About Nothing: Numbers, Hallucinations, and
Fictions. Oxford; New York: Oxford University Press.
Bär, C., Gauduchon, P. and Moroianu, A., 2005. Generalized cylinders in
semi-Riemannian and spin geometry. Mathematische Zeitschrift, 249(3),
pp.545–580. https://doi.org/10.1007/s00209-004-0718-0.
Batterman, R.W., 2010. On the explanatory role of mathematics in empirical
science. The British Journal for the Philosophy of Science, 61(1), pp.1–
25. https://doi.org/10.1093/bjps/axp018.
Bourguignon, J.-P. et al., 2015. A Spinorial Approach to Riemannian and
Conformal Geometry, EMS Monographs in Mathematics. EMS Press.
https://doi.org/10.4171/136.
Brauer, R. and Weyl, H., 1935. Spinors in n dimensions. American Journal
of Mathematics, 57(2), pp.425–449. https://doi.org/10.2307/2371218.
Button, T. and Walsh, S., 2018. Philosophy and Model Theory. Oxford Univer-
sity PressOxford. https://doi.org/10.1093/oso/9780198790396.001.0001.
Cartan, E., 1913. Les groupes projectifs qui ne laissent invariante aucune
multiplicité plane. Bulletin de la Société Mathématique de France, 41,
pp.53–96. Available at: <https:// eudml.org/ doc/ 86329> [visited on
12 January 2024].
Cartan, E., 1966. The Theory of Spinors. Cambridge, MA: MIT Press.
54 Adrian Heathcote

Chevalley, C., 1997. The Algebraic Theory of Spinors and Clifford Algebras.
Ed. by P. Cartier and C. Chevalley, Collected works / Claude Chevalley,
vol. 2. Berlin [etc.]: Springer.
Daly, C. and Langford, S., 2009. Mathematical explanation and indispens-
ability arguments. The Philosophical Quarterly, 59(237), pp.641–658.
https://doi.org/10.1111/j.1467-9213.2008.601.x.
Dedekind, R., 1888. Was sind und was sollen die Zahlen? Braunschwieg:
Friedrich Vieweg & Sohn. Available at: <http://www.digibib.tu-bs.de/
?docid=00024927> [visited on 30 October 2020].
Donaldson, S.K. and Sullivan, D.P., 1989. Quasiconformal 4-manifolds. Acta
Mathematica, 163, pp.181–252. https://doi.org/10.1007/BF02392736.
√
Estermann, T., 1975. The irrationality of 2. The Mathematical Gazette,
59(408), pp.110–110. https://doi.org/10.2307/3616647.
Euclid, 1956. The Thirteen Books of Euclid’s Elements (Book X-XIII), vol. 3
(T.L. Heath, Trans.). New York: Dover Publications.
Farmelo, G., 2009. The Strangest Man: The Hidden Life of Paul Dirac, Mystic
of the Atom. New York: Basic Books.
Field, H.H., 1980. Science Without Numbers: A Defence of Nominalism,
Library of Philosophy and Logic. Oxford: Basil Blackwell.
Fowler, D.C., 1999. The Mathematics of Plato’s Academy: A New Recon-
struction. 2nd ed. Oxford: Clarendon press.
Hahn, L.-s., 1994. Complex Numbers and Geometry, Spectrum Series. Wash-
ington, DC: Mathematical Association of America.
Heathcote, A., 2014. On the exhaustion of mathematical entities by structures.
Axiomathes, 24(2), pp.167–180. https://doi.org/10.1007/s10516-013-
9223-6.
Heathcote, A., 2021. Multiplicity and indiscernibility. Synthese, 198(9),
pp.8779–8808. https://doi.org/10.1007/s11229-020-02600-8.
Kähler, E., 2003. Il Regno delle Idee. In: R. Berndt and O. Riemenschneider,
eds. Mathematische Werke / Mathematical Works. Berlin: Walter de
Gruyter, pp.932–938.
Kalligas, P., Mpala, C., Baziotopoulou-Valavani, E. and Karasmanēs, B.,
eds., 2020. Plato’s Academy: Its Workings and Its History. Cambridge:
Cambridge University Press.
Realism, irrationality, and spinor spaces 55

Kaplansky, I., 2003. Linear Algebra and Geometry: A Second Course. Mine-
ola, N.Y: Dover Publications.
Knorr, W.R., 1975. The Evolution of the Euclidean Elements: A Study of the
Theory of Incommensurable Magnitudes and Its Significance for Early
Greek Geometry, Synthese Historical Library, vol. 15. Dordrecht: D.
Reidel.
Kocik, J., 2007. Clifford algebras and Euclid’s parametrization of Pythagorean
triples. Advances in Applied Clifford Algebras, 17(1), pp.71–93. https:
//doi.org/10.1007/s00006-006-0019-2.
La Nave, F. and Mazur, B., 2002. Reading Bombelli. The Mathematical
Intelligencer, 24(1), pp.12–21. https://doi.org/10.1007/BF03025306.
Leibniz, G.W., 1989. Dialogue on Human Freedom and the Origin of Evil
(1695). Philosophical Essays (R. Ariew and D. Garber, Trans.). Indi-
anapolis, Ind.: Hackett Publ, pp.111–116.
Lounesto, P., 2001. Clifford Algebras and Spinors. 2nd ed, London Mathe-
matical Society Lecture Note Series, 286. Cambridge, UK: Cambridge
University Press.
Malament, D., 1982. Science without numbers by Hartry H. Field. Journal of
Philosophy, 79(9), pp.523–534. https://doi.org/10.5840/jphil198279913.
Mazur, B., 2004. Imagining Numbers: (particularly the Square Root of Minus
Fifteen). 1st ed. New York: Picador.
Melia, J., 2000. Weaseling away the indispensability argument. Mind,
109(435), pp.455–480. https://doi.org/10.1093/mind/109.435.455.
Newstead, A. and Franklin, J., 2012. Indispensability Without Platonism. In:
A. Bird, B. Ellis and H. Sankey, eds. Properties, Powers, and Structures:
Issues in the Metaphysics of Realism. London: Routledge, pp.81–97.
Owens, J., 1982. Faith, ideas, illumination, and experience. The Cambridge
History of Later Medieval Philosophy. Cambridge: Cambrdige University
Press, pp.440–459. Available at: <http://opac.regesta- imperii.de/id/
252620>.
Payne, W.T., 1952. Elementary spinor theory. American Journal of Physics,
20(5), pp.253–262. https://doi.org/10.1119/1.1933190.
56 Adrian Heathcote

Penrose, R., 2000. Mathematical Physics in the 20th and 21st Centuries.
In: V.I. Arnol’d, M. Atiyah, P. Lax and B. Mazur, eds. Mathematics:
Frontiers and Perspectives. Providence: American Mathematical Society,
pp.219–234.
Penrose, R., 2004. The Road to Reality: A Complete Guide to the Laws of the
Universe. 1st american ed. Alfred A. Knopf Inc.
Penrose, R. and Rindler, W., 1987. Spinors and Space-Time. Vol. 1: Two-
Spinor Calculus and Relativistic Fields, Cambridge Monographs on
Mathematical Physics. Cambridge: Cambridge University Press.
Penrose, R. and Rindler, W., 1988. Spinors and Space-Time. Vol. 2: Spinor
and Twistor Methods in Space-Time Geometry, Cambridge Monographs
on Mathematical Physics. Cambridge: Cambridge University Press.
Renou, M.-O. et al., 2021. Quantum theory based on real numbers can be
experimentally falsified. Nature, 600(7890), pp.625–629. https://doi.org/
10.1038/s41586-021-04160-4.
Resnik, M.D., 1983. Hartry H. Field: Science without numbers. Noûs, 17(3),
p.514. https://doi.org/10.2307/2215268.
Resnik, M.D., 1985. How nominalist is Hartry Field’s nominalism? Philo-
sophical Studies, 47(2), pp.163–181. https : / / doi . org / 10 . 1007 /
BF00354144.
Shing-Tung, Y., 2000. Review of Geometry and Analysis. In: V.I. Arnol’d,
M. Atiyah, P. Lax and B. Mazur, eds. Mathematics: Frontiers and Per-
spectives. Providence: American Mathematical Society, pp.353–401.
Shiu, P., 1999. More on Estermann and Pythagoras. The Mathematical
Gazette, 83(497), pp.267–269. https://doi.org/10.2307/3619055.
Siu, M.-K., 1998. Estermann and Pythagoras. The Mathematical Gazette,
82(493), pp.92–93. https://doi.org/10.2307/3620162.
Steiner, M., 1998. The Applicability of Mathematics as a Philosophical
Problem. Cambridge, MA: Harvard Univ. Press.
Trautman, A., 1998. Pythagorean Spinors and Penrose Twistors. In:
S.A. Huggett et al., eds. The Geometric Universe: Science, Geome-
try, and the Work of Roger Penrose. Oxford: Oxford University Press,
pp.411–419. https://doi.org/10.1093/oso/9780198500599.003.0031.
Realism, irrationality, and spinor spaces 57

Van Heijenoort, J., 1967. From Frege to Gödel: A Source Book in Mathemati-
cal Logic, 1879-1931. Cambridge: Harvard University Press. Available
at: <http://archive.org/details/fromfregetogodel0000vanh> [visited on
23 January 2024].
Veblen, O., 1933. Geometry of four-component spinors. Proceedings of the
National Academy of Sciences, 19(5), pp.503–517. https://doi.org/10.
1073/pnas.19.5.503.
Veblen, O., 1934. Spinors. Science, 80(2080), pp.415–419. https://doi.org/10.
1126/science.80.2080.415.
Waterhouse, W.C., 2012. Square root as a homomorphism. The American
Mathematical Monthly, 119(3), pp.235–239. https://doi.org/10.4169/
amer.math.monthly.119.03.235.
Weyl, H., 1931. Theory of Groups and Quantum Mechanics. London:
Methuen.
Weyl, H., 1939. The Classical Groups; Their Invariants and Representa-
tions, Princeton mathematical series. Princeton, N.J.; London: Princeton
University Press; H. Milford, Oxford University Press.
Wigner, E., 1939. On unitary representations of the inhomogeneous Lorentz
group. The Annals of Mathematics (second series), 40(1), pp.149–204.
https://doi.org/10.2307/1968551.