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Sunday, 29 November 2020

A Modern Day Mathematical Platonist — Alain Badiou


 

Alain Badiou (1937-) is a French philosopher. At one point he was the chair of Philosophy at the École normale supérieure (ENS) and founder (with Michel Foucault, Gilles Deleuze and Jean-François Lyotard) of the Faculty of Philosophy at the Université de Paris VIII. He’s now René Descartes Chair and Professor of Philosophy at The European Graduate School. Badiou has also been involved in a politics and political organisations since early in his life. Indeed he has often commented on both French and global political affairs.

More relevantly to this piece, Badiou has a strong mathematical background. He’s the son of the mathematician Raymond Badiou (1905–1996). And, according to Badiou himself, by 1967 he “already had a solid grounding in mathematics and logic”. Badiou again describes his own history when he tells us that he studied “contemporary mathematics in greater depths by taking the first two years of university math”. He then goes on to say that “[t]his was from 1956 to 1958, my first two years at at the École Normale Supérieure”.

Introduction

Despite writing this piece, much — or at least some — of the philosophical things Badiou says about mathematics I (to be honest) find incomprehensible. Either that or plain pretentious. And many of the explanations (or “textual analyses”) of Badiou’s ideas by other academics are even worse than his own. I’m not alone here. Various critics have a problem with Badiou’s philosophy of mathematics. For example, the English philosopher Roger Scruton (who died in January this year) questioned Badiou’s knowledge of the foundations of mathematics. He once wrote that Badiou doesn’t

“really understand [philosophically or mathematically?] what he is talking about when he invokes (as he constantly does) Georg Cantor’s theory of transfinite cardinals, the axioms of set theory, Gödel’s incompleteness proof or Paul Cohen’s proof of the independence of the continuum hypothesis”.

Added to all that, the mathematician Alan Sokal and physicist Jean Bricmont can be quoted stating the following words:

“Badiou happily throws together politics, Lacanian psychoanalysis and mathematical set theory […] After a brief discussion on the situation of immigrant workers, Badious refers to the continuum hypothesis.”

All the above is odd if one considers Badiou’s background and grounding in mathematics (which has just been mentioned). Then again, a grounding and background in mathematics doesn’t guarantee Badiou — or anyone else — anything. It certainly doesn’t guarantee that Badiou’s philosophy of mathematics will offer us anything worthwhile — or, in his case, comprehensible.

Having said all that, some of what Badiou says about mathematics is both crystal clear and informative. Indeed obviously this piece wouldn’t have been written if that weren’t the case. However, it’s interesting to note that most of the comprehensible quotes in the following piece are from Badiou’s book In Praise of Mathematics. Now that book is actually a translation (by Susan Spitzer) of Badiou’s “dialogue” with Gilles Haéri (who’s the Director General of the publishing house Éditions Flammarion). And one can only assume that the fact that it’s both an introductory dialogue and a (perhaps liberal) English translation may help explain its comprehensibility.

Alain Badiou’s Platonism

Alain Badiou is a Platonist. Or, more accurately, he’s a mathematical Platonist. This isn’t to say that Badiou expresses his Platonism in precisely the same way in which other Platonists have done so in the past. And Badiou is certainly a considerable distance from Plato himself on many matters (which is hardly a surprise).

In any case, Badiou says that Plato is “my old master”. He also tells us about Plato’s own position on the importance of mathematics. Badiou writes:

“For Plato… mathematics was the very foundation of universal rational knowledge: the philosopher absolutely had to begin with mathematics. Even if he ultimately went beyond it, he had to learn mathematics first. Plato thought that political leaders, for example, would be well advised to study higher mathematics for at least ten years. He indicated that they were not to be satisfied with just the minimum, since they had to do geometry in space in particular.”

Badiou also explains the “Platonic” position on mathematics . He states that

“the realist (or Platonic) conception, which holds that the object of mathematics exists outside of us”.

More specifically on mathematics and its relation to the world (or to nature), Badiou says that the “Platonic” vision has it that “mathematics is part of the thinking of what there is, of what is”.

All this can be turned into a specific position on the nature of physics.

Badiou expresses a view of mathematics that’s fairly standard among the more philosophical of physicists. That is, he believes that maths “is the science of everything that is”. Indeed with mathematics everything is “grasped as its absolutely formal level”. And Badious certainly believes that mathematics can be used to describe and explain “everything that is”.

Platonism is also tied to a belief in what Badiou himself calls “absolute truths”. Having said that, Badiou does offer a very 21st-century take on such absolute truths — as can be seen in the following passage:

“I am convinced that there are absolute truths, which, although extracted at the time of their creation from a particular soil (a moment in history, a country, a language, and so on), are nevertheless constructed in such a way that their value becomes universalized.”

I’ve just used the modifying phrase “a very 21st-century take” (i.e., on absolute truth), and that certainly applies to Badiou’s sociological and historical claim that mathematical truths are

“extracted at the time of their creation from a particular soil (a moment in history, a country, a language, and so on)”.

This means that Badiou’s position can be seen as a sociological and historical account of absolute truth. And that account certainly goes against many strong — indeed Platonic — philosophical stances on absolute truth. Or, at the very least, Badiou’s account muddies the philosophical water in that although he accepts absolute truths, he still places them within various contingent contexts. (This may seem — at least to some — to be an oxymoronic position.) The basic point is that although absolute truths do indeed exist, they’re still “discovered” within very specific historical and sociological contexts. And therefore their constructions (if that’s an appropriate word here) display the contingencies of these domains.

Is this, then, simply an ornate way of saying that the equation 2 + 2 = 4 has been expressed by many different symbols and in many different languages? Indeed the same — almost exactly the same — kind of argument is made by philosophers when they talk about propositions. That is, the very same proposition can be expressed in many different languages and in many different ways (see here).

Whatever kind of Platonism Badiou’s philosophy of mathematics actually is, it’s still clear that it can be strongly distinguished from mathematical empiricism — or, indeed, from any kind of empiricism. Badiou is explicit about this when he says that he “reject[s] the theory that mathematics derives from sensory experience”. He then explains his position in a semi-Kantian manner in the following:

“[T]he real of sensory experience is thinkable only because mathematical formalism thinks, ‘ahead of time,’ the possible forms of everything that is.”

Badiou’s Pythagoreanism or Platonism?

Badiou defends his Platonic position by citing the case of the complex numbers, the imaginaries”. These were initially “invented as a pure game”. (See the later section on formalism.) That’s why they were called “imaginaries”: in order “to make it clear that they didn’t exist”. However, later they “became a basic tool used in electromagnetism in the nineteenth century, something that no one could have foreseen”.

What point is Badiou making here?

Basically, Badiou’s point is that mathematics is always found to be instantiated in the actual world (or “the real” as he puts it). Perhaps Badiou means “nearly always” as it’s certainly the case that some mathematics (or some mathematical ideas) are certainly not instantiated in nature. The physicist Roger Penrose, for example, cites various examples of this. The argument is that if mathematics and the world are one, then why are there (to use Penrose’s words) “bodies of maths with no discernible relations to the physical world”? In terms of Penrose’s actual examples, we have the following:

“Cantor’s theory of the infinite is one noteworthy example… extraordinary little of it seems to have relevance to the workings of the physical world as we know it… The same issue arises in relation to… Gödel’s famous incompleteness theorem. Also, there are the wide-ranging and deep ideas of category theory [mentioned by Badiou too — see later section] that have yet seen rather little connection with physics.”

However, is saying that mathematics is instantiated in nature (or even that maths can describe nature) the same as saying that (as Badiou does) “everything” literally is mathematical? Perhaps, then, this is (almost) a definitional truth. That is:

  1. If everything can be described by mathematics,
  2. Then everything must be mathematical.

This takes us on to the subject of Pythagoreanism.

It’s often difficult to tell whether a philosopher of mathematics is taking a Pythagorean or a Platonic position on the relation between mathematics and the world (or nature). Indeed it is sometimes difficult to distinguish the two regardless of the philosopher — or physicist! — being commented upon.

Badiou himself tells us that “mathematics touches the real”. Mathematics does so “in a way that is not experimental”. Badiou goes on to say that mathematics can’t be experimental because it is “presupposed in experience”. The ultimate reason for this is that “the real” is mathematical — or, perhaps more tellingly, already mathematical.

Yet surely if mathematics “touches the real”, then it can’t actually be “the real”. That is, if a finger touches a flower, then the finger isn’t the actual flower it touches. Perhaps this is simply a problem brought about by Badiou’s metaphor “touches”. Yet that metaphor (if it is a metaphor) does at least hint at a separation between mathematics and the world (or the real). That’s unless mathematics is touching itself! In that case, do we have the following Pythagorean identity? -

The Real = mathematics
(The Slovenian philosopher Slavoj Žižek always capitalises the word “real” — à la Hegel — and puts the definite article before it.)

Badiou then goes on to argue that

“even the great instruments that are used in experiments, from telescopes to giant particle accelerators, are ‘materialized theory,’ and presuppose, even in the way they’re constructed, extremely complex mathematical formalisms”.

This reads like a reworking of the Pythagorean phrase “all things are numbers”. That is, mathematics is immanent in the world. Indeed mathematics is even immanent in the instruments which manipulate that world. To put it plainly, this isn’t the case of mathematics describing the world. This is a case of mathematics — or numbers — literally being instantiated in the world (or, in this particular case, in instruments).

Yet even here we can question any commitment to the world actually being mathematical. After all, Badiou uses the phrase “materialized theory”. Doesn’t that hint at a separation between mathematical theory and its materialization? That is, firstly there is the (mathematical) theory, and only then is that theory materialized. Thus the mathematics must surely antedate its materialization (in this case, in instruments).

This situation is not unlike Plato’s ante rem position on universals in which universals exist before they are instantiated by (or in) particulars. Aristotle, on the other hand, believed that universals exist post rem — i.e., only after they’re instantiated by (or in) particulars. So now it can now be said that just as there’s a separation between a universal and its instantiation (or exemplification), so there’s also a separation between a mathematical theory and its instantiation in the guise of a instrument — or its instantiation in anything else in the world. Thus if the mathematics antedates (“its”) instantiations (or concretisations), then doesn’t that call into question the idea that the world (or its parts) is (or are) literally mathematical? The most we can say is that the world can embody mathematical (to use Badiou’s term) “formalisms” — not that it is mathematics.

It’s true that instruments (to stick with Badiou’s own example) “presuppose” the mathematics in that they (metaphorically at least) abide by the mathematics. Or, more correctly, mathematics describes the instruments and the instruments must also adhere to mathematical formalisms. But simply because these instruments — and all the parts of the natural world — can be described by mathematics, that doesn’t mean that the world literally is mathematics. And the metaphorical word “abide” (or “adhere”) may not help in that the instruments must have mathematical properties (or properties that can be described by mathematics) in order for them to be what they are. However, almost all things can be described by maths — even random or chaotic events. Yet that still doesn’t mean that they actually are mathematical.

Platonic Structuralism

Alain Badiou is explicit about his structuralism — specifically when it comes to mathematics. He writes:

“Structures are first and foremost the business of mathematicians.”

He then cites his influence which not many mathematical structuralists will recognise. In more direct terms, Badiou appears to suggest that his mathematical structuralism came via his interest in — and knowledge of — Claude Lévi-Strauss’s anthropological structuralism. Badiou tells us that

“[a]t the very end of [Levi-Strauss’s] seminal book, The Elementary Structures of Kinship, the great anthropologist [] referred to the mathematician Weil to show the exchange of women could be understood by using the algebraic theory of groups”.

So the interesting thing about Badiou’s mathematical Platonism is that it’s a Platonism about structures, not numbers.

In terms of what Badiou calls “being”, the only way to to capture “being” is “to think purely formal structures”.

So what are “formal structures”?

They’re “structures indeterminate as their physical characteristics”. How does this connect to mathematics? It does so because “the science of these structures is mathematics”.

Badiou also cites a specific example of mathematical structuralism from the world of mathematics: category theory. Badiou tells us category theory

“is roughly the theory of relations ‘in general,’ with no pre-specification of given objects”.

Badiou then goes into more detail when he refers to the nature of category theory. Here he emphasises relations rather than structures. Indeed it can be seen that relations and structures are intimately tied together. Badiou tells us that

“a structural edifice is gradually built up in which relations seem to prevail over entities, or objects, or even to determine their nature and properties”.

Badiou then makes the obvious conclusion:

“So it is tempting to reduce all the so-called ‘intuitive’ objects to structural, or formal, manipulations whose principles only objects the mathematician’s decisions or choices. What then ‘exists’ are structured domains, which are accountable only to the formalism by which they are exhibited.”

The problem here is that Badiou is against what he calls “formalism”. Yet it’s the formalism that brings about the structures and relations which Badiou believes trump mathematical “entities, or objects”. That is, numbers are the children of the structures and relations and therefore of the formalisms which bring about the structures and relations.

Badiou then gives a more concrete and everyday example of mathematical structuralism.

He concentrates on the word “successor”. Firstly Badiou tells us that “[m]athematical thinking makes a tentative appearance if you say that 235,678,982 is the ‘successor’ of the number 235,678,981”. More relevantly:

“But you can then see that what really matters is the word ‘successor,’ which actually denotes an operation and therefore, ultimately, a structure, in this case addition: if the number n exists, whatever n may be, then there also exists the number n + 1, which will be called the successor of n.”

The English cosmologist and mathematician John D. Barrow also makes similar points (though he stresses operation, not structure). He writes:

“This is done by focusing attention upon the operation by which numbers are changed rather than upon the numbers themselves. Thus, a simple counting process like 1, 2, 3 … is seen not as a list of particular numbers but as the result of carrying out a particular operation of change upon a number, thereby generating its successor.”

Both the quoted passages directly above are slightly problematic in that even if numbers are generated by a structure (in this case addition), then this very structure also begins with a number — even if only an unspecified n. In other words, the operations and nature of the successors — and therefore the structure itself — seem to be dependent on the “existence” of n (which is an unspecified number). Even though n is unspecified and Badiou uses the words “carrying out a particular operation of change upon a number”, it’s still a number (or something which is taken to be a number) that begins the show: that is, not a structure or an operation.

It can of course be argued that n itself is a product of prior structures. But that simply replicates the problem in that these prior structures might also have depended on the existence of prior numbers. This means that the numbers (on this reading at least) generate the structures, rather than vice versa.

It may be concluded, then, that the reality of succession and addition (as well as of relations and structures) are distinguishable from the reality of numbers. That means that the successor relation and the operation of addition (as well as structures and relations — at least in mathematics) all depend on the prior existence of numbers. In other words, the prior nature of numbers generate the nature of the structures. Consequently, why can’t we make, say, the number 2 have a prior nature which is completely separate from it having a place in a structure — or, indeed, from it having any relations at all to other numbers?… Having said all that, this position hardly makes sense either!

Against Formalism

As many Platonists have done, Badiou pits himself against what he and many others have called “formalist” accounts of mathematics. Indeed his account of mathematical formalism is also — at least in part — an account of Ludwig Wittgenstein’s ostensibly extreme constructivism when it came to mathematics (see here).

Badiou’s fundamental point is that mathematics isn’t “purely and simply a formal, arbitrary game”. He claims that the formalist simply desires a

“codification of a language that is of course formally rigorous, since the concepts of deduction and proof are normative and formalized in it, but whose rigor cannot claim to have an ongoing relationship with empirical reality”.

In other words, maths in neither about the world nor derived from the world. One reason for this is that formalists believe (at least according to Badiou) that “[m]athematical axioms can be changed, after all”. It is, therefore, the axioms (along with the logical rules) that (as it were) create their own formal worlds.

Badiou, on the other hand, believed that maths has “content”. He says that “there is a real ‘content’ to mathematical thought”. Badiou continues:

“[Maths is] neither a language game — even if complex formalisms are required — nor is it an offshoot of pure logic.”

Indeed Badiou believes that “the majority of mathematicians” take this position too.

The mathematical physicist and mathematician Roger Penrose also replicates Badiou’s position on mathematical formalism. Penrose writes:

“The point of view that one can dispense with the meanings of mathematical statements, regarding them as nothing but strings of symbols in some mathematical system, is the mathematical standpoint of formalism.”

Penrose has a serious problem with the “point of view”. He goes on to say that “[s]ome people like this idea, whereby mathematics becomes a kind of ‘meaningless game’”. Penrose concludes:

“It is not an idea that appeals to me, however. It is indeed ‘meaning’ — not blind algorithmic computation — that gives mathematics its substance. Fortunately, Gödel dealt formalism a devastating blow!”

Perhaps Penrose’s quote above sums up what it is that Alain Badiou’s own mathematical Platonism is all about.

Thursday, 19 November 2020

Gödel’s First Incompleteness Theorem in Simple Symbols and Simple Terms


 


The following piece explains a particular symbolic expression (or version) of Kurt Gödel’s first incompleteness theorem. It also includes a particular expression (or example) of a Gödel sentence (i.e., “This statement is false” — this link takes you to a humorous entry!). In terms of the actual symbols used, this representation of the theorem expresses a (slight) philosophical and logical bias. So it’s worth noting that almost every symbolisation of the theorem is unique — if sometimes only in tiny detail. (In logic and mathematical logic there’s the common phenomenon of various symbolic “dialects” competing with — or simply complementing — each other.) And this representation and explanation exclude all the other details which surround the bare theorem itself. Indeed this symbolic representation alone doesn’t prove or demonstrate anything. And even when the symbols are defined or interpreted, that’s still the case. In addition, it’s worth distinguishing the truth (and lack of proof ) of a Gödel sentence from any proof of the first incompleteness theorem itself — even if the two can’t be entirely disentangled!

Thus the following words don’t attempt to tackle the arguments and extra details which are required to establish the theorem. And neither do they extrapolate anything from it. However, even this basic approach is bound to leave out much detail. And that’s simply because this is a short introduction to a particular symbolisation of Gödel’s first incompleteness theorem.


Three things need to be noted to begin with:

  1. The first incompleteness theorem is essentially about systems and the truth-values of certain statements within those systems. (Alternatively, the first incompleteness theorem is about a particular system and a Gödel sentence within that particular system.)
  2. Those systems and statements are arithmetical and therefore use natural numbers. (In other words, the first incompleteness theorem is not applied across the board — as it often is.)
  3. Within those systems there are some true statements about natural numbers which cannot be proved within those systems. (Alternatively, within a given system there will be a true statement about natural numbers that cannot be proved within that system.)

To get to the core of Kurt Gödel’s first incompleteness theorem, let’s sum it up in its bare logical (or symbolic) form. This particular symbolism (just one among many) will hopefully capture what’s at the heart of the theorem.

Take the following symbolic representation from the logician and philosopher Professor Alasdair Urquhart (as found in his paper ‘Metatheory’):

G ↔ ¬Prov(G⌝)

The following is a list of definitions of the symbols in the biconditional theorem above:

G = a Gödel sentence
= if and only if (i.e., the biconditional symbol)
¬ = negation (or “not”)
Prov = provable
¬Prov = not provable
G = The “code number” of the Gödel sentence G. (The superscripted Quine corners are — basically — quotation marks.)

Thus G ¬Prov (⌜G⌝ ) means:

The sentence “This sentence is false” is true if and only if it is not provable in system T (i.e., the system to which it belongs).

Or:

Gödel sentence G is true if and only if there is no proof of G in system T (i.e., the system to which it belongs).

So why is the symbol G put in brackets after the if and only if (i.e., ↔) sign and the sign (i.e., ¬) for negation? Why do we have the symbol ⌜ G rather than plain G? This is because the brackets (i.e., ⌜ and ⌝) symbolise self-reference or “quotation”. That is, firstly we have the symbol G, and then when we refer to G we get ⌜G ⌝.

Thus ⌜ G⌝ is a “code number”.

A code number is a number which is used to identify something. This means that ⌜ G⌝ is the code number of the Gödel sentence G (i.e., the symbol G without brackets). Furthermore, a Gödel number is a specific kind of code number. In mathematical logic, Gödel numbers are natural numbers which are assigned to statements (as well as to the individual symbols within those statements ) within a given system or formal language.

In terms of the biconditional symbol (i.e., ↔).

This symbolises that both sides of the equality sign (i.e., =) are logically equivalent in that both are either jointly true or jointly false. Note: this doesn’t also mean that they have the same meaning.

This is one expression of the aforesaid biconditional:

i) G 
is true
if and only if 
ii) ¬Prov(⌜
G⌝) 
is true.

Alternatively, the inversion (i.e., since the theorem includes a biconditional):

i) ¬Prov(⌜ G⌝)
is true
if and only if
ii)
G
is true.

The Gödel sentence G (in this instance, “This statement is false”) is self-referential. That is, it refers to itself (or G refers to G). The archetype of this Gödel sentence is the Liar paradox; which is also self-referential. Indeed self-reference is at the heart of the whole show! Without self-reference we wouldn’t have a Gödel sentence or the problems and insights which arise from it. (See my ‘Why Empty Logic Leads to the Liar Paradox’.)

In addition, Gödel sentence G is true if and only if there is no proof of G. Ordinarily it’s taken that a mathematical statement P is taken to be true if and only if there is a proof of P. Gödel’s first incompleteness theorem is saying the literal opposite of that.

What’s also important here is to note the Gödel sentence’s position in a system (or theory). None of this makes any sense outside the context of the system (or theory) to which Gödel sentence G belongs. In other words, taking G entirely on its own makes no sense at all.

Tuesday, 17 November 2020

Is Roger Penrose a Platonist or a Pythagorean?


 

Roger Penrose is not only a mathematical physicist: he’s also a pure mathematician. So it’s not a surprise that Penrose expresses the deep relation between mathematics and the world (or nature) in the following way:

“[T]he more deeply we probe the fundamentals of physical behaviour, the more that it is very precisely controlled by mathematics.”

What’s more:

“[T]he mathematics that we find is not just of a direct calculational nature; it is of a profoundly sophisticated character, where there is subtlety and beauty of a kind that is not to be seen in the mathematics that is relevant to physics at a less fundamental level.”

Penrose is (rather obviously) profoundly aware of the importance of mathematics to (all) physics. Yet, more relevantly to this piece, he’s also aware that maths alone can sometimes (or often) lead the way in physics… and sometimes in a negative manner! So despite the eulogies to mathematics above, Penrose offers us these words of warning:

“In accordance with this, progress towards a deeper physical understanding, if it is not able to be guided in detail by experiment, must rely more and more heavily on an ability to appreciate the physical relevance and depth of the mathematics, and to ‘sniff out’ the appropriate ideas by use of a profoundly sensitive aesthetic mathematical appreciation.”

Platonism and Pythagoreanism in Contemporary Physics

Roger Penrose is a Platonist, not a Pythagorean. (Or at least he’s a Platonist in certain respects — see here, here and my ‘Platonist Roger Penrose Sees Mathematical Truths’ ) One reason why this can be argued is that Penrose admits that he

“might baulk at actually attempting to identify physical reality within the reality of Plato’s world”.

To the Pythagorean, the world literally is mathematical. Or, perhaps more accurately, the world literally is mathematics (i.e., the world is literally constituted by numbers, equations, etc.). That may sound odd. However, if we simply say that “the world is mathematical”, then that may (or does) only mean that the world can be accurately — even if very accurately — described by mathematics. The Pythagorean, however, states such phrases as “things are numbers”. He therefore establishes a literal identity between maths and the world (or parts thereof).

To the Platonist, on the other hand, the mathematical world is abstract and not at all the same as “physical reality”. (Plato often actively encouraged philosophers and mathematicians to turn their eyes — or souls — away from the physical world.) Yet it’s still undoubtedly the case that abstract mathematics — even Platonic mathematics — is a fantastic means to describe the world. Despite that, the Platonic world is still abstract and not identical to the physical world. In other words, there is no identity between the physical world and the Platonic world. However, there is an identity between the physical world and the Pythagorean world.

More generally, even a (at times) hard-headed positivist (see here) like Werner Heisenberg recognised the importance of the Pythagorean tradition in physics. He argued that

“this mode of observing nature, which led in part to a true dominion over natural forces and thus contributes decisively to the development of humanity, in an unforeseen manner vindicated the Pythagorean faith”.

All that may depend on what Heisenberg meant by the word “Pythagorean”. After all, it’s often the case that the word “Pythagorean” is simply used as a literal synonym for the word “Platonic”. Thus having said all the above, such distinctions between Platonism and Pythagoreanism (at least in these specific respects) may be a little vague or even artificial.

This may apply to Roger Penrose’s position too.

Take Penrose’s own (as it were) quasi-Pythagorean reading of the “complex-number system”. He writes:

“Yet we shall find that complex numbers, as much as reals, and perhaps even more so, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.”

Since the passage above is fairly poetic, it’s difficult to grasp exactly how Pythagorean it actually is. More clearly, surely the words

“[i]t is as though Nature herself is as impressed by the scope and consistency of the complex-number system”

are purely poetic — even if there’s a non-poetic “base” that’s expressed by the poeticisms. Penrose does, after all, prefix the statement above with the words “[i]t is as though”. So surely it can be said that Nature doesn’t need (or require) the complex-number system. It is us human beings (or physicists) who need that system in order to describe Nature.

In any case, Penrose’s strongest (possibly Pythagorean) claim is:

The complex numbers “find a unity with nature”.

Now is that “unity” also an identity? Not necessarily. After all, numbers may be united with Nature only in the sense that they can describe it perfectly. Saying that numbers are identical with nature, on the other hand, is something else entirely. As it is, the phrase “unity with nature” is hard to untangle. (Hence my use of the word “poetic” earlier.)

Having put a quasi-Pythagorean position on (at the least) complex numbers, Penrose then puts a (literally) down-to-earth position on the real numbers. Penrose writes:

“Presumably this suspicion arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians.”

Despite using the phrase “down-to-earth position” before the quote above, this passage is at least partly Pythagorean in that it states that

“distances, times, and other physical quantities [] provid[ed] the reality” which real numbers “required”.

This can be read as meaning that the real numbers are (as it were… or not) embodied in distances, times and other physical quantities. Yet — historically at least — it seems that complex numbers didn’t pass that Pythagorean test.

Examples: Paul Dirac, Etc.

It’s undoubtedly the case that various well-known (as well as largely unknown) physicists have often been led by mathematics when it comes to their theories. That is, they certainly haven’t always been led by experiments or by observation.

Take the case of Paul Dirac.

Dirac found the equation for the electron (see here). He also predicted the electron’s anti-particle (see here). Both the finding and the prediction came before any experimental evidence whatsoever.

Penrose calls Dirac’s finding of the equation for the electron an “aesthetic leap”. However, Penrose also says that it arose

“from the sound body of mathematical understanding that had arisen from the experimental findings of quantum mechanics”.

That basically means that although Dirac’s mathematics was (as it were again) pure, “the experimental findings of quantum mechanics” must still have been swirling around in Dirac’s head as he carried out his pure mathematics.

The Dirac case also shows us the to and thro between (pure) maths and experimental findings. That is, even if we have aesthetic and/or mathematical leaps, the mathematical physicists concerned were clearly still aware of the experimental findings which proceeded their abstract leaps. What’s more, Dirac’s own leaps were “made with great caution and subsequently confirmed in observation”. Indeed in both Dirac’s cases, confirmation came very quickly.

A purely philosophical slant can be put on the Dirac case. (Although I’m a little wary of shoehorning philosophical terms — or ways of thinking - onto what physicists have done.) As the philosopher James Ladyman (technically) puts it:

“Sophisticated inductivism is not refuted by those episodes in the history of science where a theory was proposed before the data were on hand to test it let alone suggest it... Theories may be produced by any means necessary but then their degree of confirmation is a relationship between them and the evidence and is independent of how they were produced.”

We can now say that in Dirac’s case there was no “data [] on hand to test it let alone suggest it”. Actually, the last clause (“let alone suggest it”) may be a little strong in that previous experiments in (quantum) physics must surely have suggested various things to Dirac. The thing is, Dirac still had no (hard) data to back up his prediction or equation. Despite that, Dirac’s theories were “produced by any means necessary” (or by any mathematical means necessary) and only then were they confirmed.

To get back to Penrose.

Penrose goes into more detail elsewhere when he says that in the cases of Dirac’s equation for the electron, Einstein’s general relativity and “the general framework” of quantum mechanics,

“physical considerations — ultimately observational ones — have provided the overriding criteria for acceptance”.

Opposed to that, Penrose goes on to say that

“[i]n many of the modern ideas for fundamentality advancing our understanding of the laws of the universe, adequate physical criteria — i.e. experimental data, or even the possibility of experimental investigation — are not available”.

Penrose then concludes by saying that

“we may question whether the accessible mathematical desiderata are sufficient to enable us to estimate the chances of success of these ideas”.

All above shows us that Penrose is still acknowledging that (in a basic sense at least) the mathematics comes first. That is, Penrose believes that any “acceptance” of the “ideas” for “our understanding of the laws of nature” often comes after the (pure) mathematics. (That’s if the maths is ever truly pure in that previous experiments, observations, physical theories, etc. will — or may — be swilling around in the mathematical physicist’s head.) To repeat: the mathematical speculation (or theorising) comes first, and only then do physicists expect the “physical considerations” to provide the “overriding criteria for acceptance”.

When it comes to many (or some) “modern ideas” (Penrose mainly has string/M theory in mind — see here), on the other hand, “physical criteria” are “not available”. Yet that was also true — as we’ve seen — of the examples which Penrose himself cites (i.e., quantum mechanics, general relativity and Dirac’s equation for the electron). In these example, physical criteria were not available at the times these ideas were first formulated. This means that the observations, confirmations, experiments, etc. came after — even if very soon after.

So what if the experiments haven’t been done? Which precise experiments must guide the physicist? And what if there are no currently relevant or possible experiments which can guide the theoretical physicist? Of course it can now be argued that if there are no relevant, actual or possible experiments (or observations), then in what sense is any given physicist — even if mathematical physicist — doing physics at all?

String Theory and Penrose’s Twistor Theory

Despite Penrose’s emphasis on the fundamentally important role of maths in physics (which is hardly an original emphasis), Penrose is still highly suspicious of the nature of string theory.

Although Penrose doesn’t always name names, he still stresses “the mathematics that is relevant to physics”. He warns that

“if it is not able to be guided in detail by experiment, [it] must rely more and more heavily on an ability to appreciate the physical relevance and depth of the mathematics, and to ‘sniff out’ the appropriate ideas by use of a profoundly sensitive aesthetic mathematical appreciation”.

This squares with what British science writer and astrophysicist John Gribbin has to say.

Gribbin too talks in terms of what he calls a “physical model” of “mathematical concepts”. He writes (in his Schrodinger’s Kittens and the Search for Reality) that “a strong operational axiom” tells us that

“literally every version of mathematical concepts has a physical model somewhere, and the clever physicist should be advised to deliberately and routinely seek out, as part of his activity, physical models of already discovered mathematical structures”.

Yet even in Gribbin’s case, it’s still clear that a “mathematical concept” comes first and only then is a “physical model” found to square with it.

Penrose’s words on his own twistor theory are also very relevant here in that after criticising string theorists for seemingly divorcing their mathematics from experiment, prediction, observation, etc., he then freely confesses that he’s — at least partly - guilty of exactly the same sin.

Firstly, Penrose tells us about the pure mathematics of twistor theory. He writes:

“Yet twistor theory, like string theory, has had a significant influence on pure mathematics, and this has been regarded as one of its greatest strengths.”

Penrose then cites a couple of very-specific examples:

“Twistor theory has had an important impact on the theory of integrable systems [] on representation theory, and on differential geometry.”

And then we have the mathematical aesthetics of twistor theory:

“Twistor theory has been greatly guided by considerations of mathematical elegance and interest, and its gains much of its strength from its rigorous and fruitful mathematical structure.”

Finally, the confession:

“That is all very well, the candid reader might be inclined to remark with some justification, but did I not complain [] that a weakness of string theory was that it was largely mathematically driven, with too little guidance coming from the nature of the physical world? In some respects this is a valid criticism of twistor theory also. There is certainly no hard reason, coming from modern observational data, to force us into a belief that twistor theory provides the route that modern physics should follow… The main criticism that can be levelled at twistor theory, as of now, is that it is not really a physical theory. It certainly makes no unambiguous physical predictions.”

So how does Penrose extract himself from this problem? Well, to be honest, he doesn’t go into great detail — at least not after these specific passages.

The obvious question to ask now is this:

What is twistor theory doing right that string theory is doing wrong?

Is the answer to that question entirely determined by how close each theory is to “the nature of the physical world”? But don’t we (as it were) get to the physical world only through theory? As Stephen Hawking once put it:

“If what we regards as real depends on our theory, how can we make reality the basis of our philosophy? But we cannot distinguish what is real about the universe without a theory… Beyond that it makes no sense to ask if it corresponds to reality, because we do not know what reality is independent of theory.”

In any case, perhaps it’s the case that (as Penrose may believe) the mathematics of twistor theory is superior to the mathematics of string theory.

String theory particularly has been criticised for not making “unambiguous physical predictions”. Yet here’s Penrose saying exactly the same thing about his own twistor theory.

Finally, there probably never is (to use Penrose’s own words) “a hard reason” to “force” us to believe any physical theory — at least not in the early days of such theories. This obliquely brings on board the largely philosophical idea of the underdetermination of theory by data in that the “modern observational data” which Penrose mentions will never be enough to force the issue of which theory to accept. In other words, whatever observational data there is can be interpreted (or theorised about) in many ways. Alternatively, the same observational data can produce — or be explained by — numerous (often rival) physical theories.