Barry Stroud’s Critique of Naturalised Epistemology

It may be worth putting Barry Stroud’s (1935–2019) critique of naturalised epistemology in some kind of context. That’s partly because this is something that Stroud himself did when he tackled philosophical problems and issues.

So here are a couple of passages from his obituary — as published in Berkeley News (a publication of the University of California):

“While best known for his work in epistemology and philosophical skepticism… Stroud’s overarching legacy, his colleagues say, was his ability to see the big picture and get to the heart of philosophy…

“As a philosopher, Stroud came of age during a time when the prevailing Western attitude was that philosophical questions could be answered by the natural or social sciences, and he challenged those ideas...

“‘One might say that, while everyone else was philosophizing about consciousness, reality and knowledge, he was philosophizing about philosophizing itself,” [Kolodny] added.

“‘Barry single-handedly brought philosophical skepticism — which gives reasons to doubt whether we can know even the most ordinary things about the world around us — back to the center of philosophical discussion,’ Bridges said.”

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In his ‘The Significance of Naturalised Epistemology’ (1981), the Canadian philosopher Barry Stroud offers a realist point against W.V.0. Quine’s naturalist position on epistemology. He makes the simple distinction between

1) Beliefs which are “constructions or projections from [sensory] stimulations” [Quine’s position].

and

2) Beliefs about the world.

Stroud’s question is: How do we get from 1) to 2)? Or:

How do we get from sensory stimulations — which are (it is assumed) caused by the world — to truths about the world or representations which are taken to be true of the world?

This is the ancient sceptical question. Indeed, at least since Descartes, this was once the central question of epistemology.

Another way of putting all this is in terms of what and how we know.

On Quine’s account, all we know about is are sensory stimulations and what we assert in response to them. This means that we don’t know anything about the world itself. (A locution which Quine would, no doubt, have rejected.) Or, as Stroud puts it, from this

“we would not in addition have independent access to the world they are about on the basis of which we could determine whether they are true”.

If what Stroud is saying is correct, then we can’t even say that sensory stimulations — and the causally resultant assertions — are “about the world” if they aren’t true of the world (or give us knowledge of the world) in the first place.

So are beliefs and assertions about our own and other people’s beliefs and assertions not directly (or even indirectly) about the world? These beliefs and assertions (or Quine’s “projections”)

“could not be seen as a source of independent information about the world against which their own truth or the truth of the earlier beliefs could be checked”.

This is as strong a statement of the possibility of scepticism in epistemology as you could hear from a sceptic himself. That is, we can compare beliefs against beliefs; though we can’t compare beliefs (or their contents) with the world (or its parts) itself. The American philosopher Donald Davidson (1917–2003) would even argue that we can’t compare beliefs, etc. with sensory stimulations because there are no belief-free sensory stimulations “which could count as evidence” in the first place.

All this can be (partly) boiled down — or is analogous — to Bishop Berkeley’s well-known statement that an “idea” can only be like another idea and not like what it’s an idea of in the world. Stroud, then, is referring to the traditional epistemological problem of “bridging the gap between sense data and bodies” (Quine, quoted in Stroud). However, whereas traditional epistemologists saw this as a problem, the logical positivists saw it is a “pseudo-problem”. Or, as Quine put it, this move between sense-data (or Quine’s “sensory stimulations”) and bodies is “‘real but wrongly viewed’” (Quine, quoted in Stroud).

In terms of this (logical) gap between “data and bodies”, Quine neither saw it as a pseudo-problem nor saw it, as Kant did, as the greatest failure of philosophy. Quine’s

“positive account does not try to show how we rule out the possibility that the world is completely different in general from the way our sensory impacts and our internal makeup lead us to think of it”.

If Quine did believe this, then we can argue that he might as well have believed that the problem of the external world (or even its existence) is a pseudo-problem if the sceptic or realist doesn’t “try to show how we rule out the possibility” that the world may be different to what we think it is. Perhaps, then, there’s a logical gap between data (or evidence) and the world. If there is, then why didn’t Quine think that such problems are pseudo-problems as the logical positivists did? Or as Stroud puts the logical positivist (or verificationist) position:

“The traditional epistemological question of the reality of the external world and our knowledge of it was for Carnap and Schlick and other verificationists a meaningless pseudo-question; no answer to it was empirically confirmable or disconfirmable.”

We can’t get between our evidence (or data) to get directly to — or at — the world in order to match the evidence (or data) with the world. Hence the logical gap. Yet physicists - or at least many of them - had always accepted that they must rely on evidence (or “phenomena” in the case of Kantian scientists like Einstein and Mach in the early 20th century). And because physics had the last (as well as first?) word on the world or nature, and if physicists happily accepted the importance of sense-data (or Carnap’s “cross-sections of experience”) when it came to world-talk, then the logical positivists did so too. Thus, as a consequence of this “deference” (Quine’s term) to physics, Stroud comments that

“[f]or Carnap we must distinguish a philosophical (pseudo) employment of a form of words from an ordinary or scientific employment of the same words”.

We mustn’t talk about the world or reality (its nature or existence) in the way the sceptics or epistemologists do. Instead we must speak as physicists (or even laypersons) speak. However, if we take the former option, then we’ll basically be talking rubbish.

In terms of Quine’s position again.

What’s the point of accepting the possibility that the “world [could be] completely different” if Quine didn’t offer us a way out of this problem? Again, if the gap is logical, then he must surely have seen the problem as a pseudo-problem. However, if all we have are “sensory impacts” and a largely given “internal makeup”, then we must surely see why Quine took the (pragmatic?) position which he did take.

Despite Donald Davidson’s criticisms of Quine’s emphasis (or very position) on sensory stimulations (hinted at earlier), Stroud argued that Quine’s position isn’t like that of the sense-data theorists (or British Empiricists/ phenomenalists). What Quine doesn’t do, even with his sensory stimulations, is try

“to isolate a domain of pure sensory data evidentially or epistemically prior to the knowledge of nature that is to be explained”.

So perhaps Quine’s position was midway between the “atomism” of sense-data theorists and Davidson’s holism. Thus:

(1) The sense-data theorists’ position: Sense-data are untouched by belief and theory.

(2) Davidson’s position: Beliefs are (clearly) touched by (other) beliefs and theory.

And , perhaps, the happy medium:

(3) Quine’s position: Sensory stimulations are touched by belief and theory but which nevertheless must ground (future) beliefs and theories.

Stroud’s perspective seems to depend on accepting a very controversial theory of metaphysically-realist truth. (Putnam once argued that, in terms of scientific truth, Quine was himself a metaphysical realist who, for example, accepted the principle of bivalence for the statements of physics— see here.)

Stroud puts this realist position by — once again — questioning Quine’s exclusive reliance on sensory stimulations and the resultant “projections” (or “posits”) we make “about the world” because of such sensory stimulations. He asked Quine this simple question:

“[How do the] subject’s ‘projections’ or ‘posits’ turn out to be correct, and not just a question about how he comes to make them [?]”

If the (to use Richard Rorty’s phrase) “world is well lost” (or if we don’t have direct access to the world), then how do we know which projections (or posits) are correct and which are incorrect without the (as it were) world telling us so? (Rorty and Davidson would say that the world can’t tell us anything — not even metaphorically or indirectly.) How does the Quinian decide which posits (or projections) are correct and which ones are incorrect? Are these decisions made exclusively on pragmatic or instrumentalist lines?

Of course Quinians can’t only be concerned with “how he comes to make” these projections (or posits) because some people come to make such projections about goblins or the influence of ley lines. So there’s more to the Quinian story than (mere) projections or posits. And, according to Stroud. that something extra is causality (or causation),

Stroud puts this rather simple causal approach to knowledge this way. He says that we

“would see that the world around [the investigator or epistemologist] is generally speaking exactly the way he says it is and that its being that way is partly responsible for his saying and believing what he does about it”.

This is certainly largely Davidson’s position and also the reason why he argued that “most of the beliefs in a coherent set of beliefs are true”. That is, we wouldn’t say (or believe) what we do about the world if the world wasn’t (as it were) responsible for what we say (or believe) about it. That relation (or connection) between belief and the world is largely accounted for in terms of causation. (According to Davidson, “causation does not come under a description” and it’s not in itself “explanatory”.) This causal (for want of a better word) position may seem simple and even a little naïve. Stroud writes that

“[m]any philosophers nowadays would hold that that is enough for knowledge: the subject believes that p, he is right, and it is no accident that he is right”.

So say that Jeff believes that P because the world is as he says it is. That is, the world causes him to believe (or say) that P in a causal-kinda-way. Indeed this almost has the appearance of being some kind of isomorphic relation between the world and what Jeff says (or believes) about it. It’s no surprise, then, that Stroud concludes by saying that the “adequacy of any such ‘causal’ account of knowledge is still questionable at best”.

And because of everything that’s just been said about Stroud’s account (i.e., that we essentially loose the world on Quine’s alternative), then we must also accept “that countless ‘theories’ could be ‘projected’ from the sensory impacts we receive”. Yet that’s no surprise at all because Quine himself admitted that. Indeed Quine was well known for stating the following: All theory is underdetermined by the sensory evidence. (All this is also part of the story of ontological relativity, the indeterminacy of meaning and the inscrutability of reference.) And that’s precisely why Quine also believed that we must employ “pragmatic” requirements and judgements when it comes to theory choice.

Stroud puts all the above less positively. He argued that because of this theory-pluralism (or theory-liberalism),

“if we do happen to accept one such ‘theory’ it could not be because of any objectively discoverable superiority it enjoys over it competitors”.

The theory we choose won’t be a truer account of the world. It won’t give us (so to speak) more of the world. It will only be pragmatically (or instrumentally) superior (to us) than the other theories. It won’t be truer or even more correct. It won’t have been chosen because something has been (as Stroud puts it) “objectively discovered” which places it in a superior position — i.e., in terms of truth rather than (mere) pragmatic utility (or whatever). That said, according to Stroud, Quine did accept an objective component to his alternative position. Yet that objective component is only the “meagre [sensory] data” which isn’t itself the world (as well as not really data or evidence on Davidson’s position). Even this pseudo-objective component doesn’t amount to much because (as already stated) this same objective data (or evidence) can be used to construct many competing, complementary or — sometimes - even contradictory theories.

Reference

Stroud, Barry, ‘The Significance of Naturalised Epistemology’ (1981)

[I can be found on Twitter here.]

 

Daniel Dennett’s Crude Ad Hominems — As Found in His Book, ‘Intuition Pumps’

Some academics (or professional) philosophers have a problem with the American philosopher Daniel Dennett (1942-) because he’s a successful writer of what’s called “popular philosophy”. (Note: The English philosopher Timothy Williamson makes a distinction between “popular philosophy” and “populist philosophy” — see here.) I don’t. After all, although Dennett has written many popular books on philosophical subjects, many of his arguments can also be found in his academic (i.e., technical) papers. Still, it’s true that as time has gone by, Dennett has written less and less academic stuff.

Is that automatically a bad thing?

Having said all that, in Dennett’s book, Intuition Pumps and Other Tools for Thinking, some of his ad hominems against his detractors are terrible: i.e., really gross and crude. So, yes, I was a little surprised by the many snide comments Dennett uses against his philosophical opponents.

Ad-Hominem Passages From Daniel Dennett

Here are five examples:

(1) Dennett: “‘I just can’t conceive of a conscious robot!’ Nonsense, I replied. What you mean is that you won’t conceive of a conscious robot.”

Response: Even if there is an element of truth in what Dennett says above, he must still know that not all people are entirely — or even at all — driven by their emotional reaction to this issue. That said, some — perhaps many — people are indeed offended — or depressed - by the very notion of a “conscious robot”.

Of course it may well also be the case that Dennett himself has an emotional reaction to this issue. That is, perhaps Dennett “won’t conceive” of the possibility that a “conscious robot” is impossible… or at least highly unlikely.

And when did Dennett become a mind-reader?

For a verificationist and/or neo-behaviourist, he seems to be very good at reading those contents of other people’s minds — those contents which most certainly haven’t been expressed in verbal behaviour.

(2) Dennett: “We found his [John Searle’s] though experiment fascinating because it was, on the one hand, so clearly fallacious and misleading argument, yet, on the other hand, just as clearly a tremendous crowd-pleaser and persuader.”

Response: Dennett surely shouldn’t claim that the Chinese room argument is “clearly fallacious and misleading”. It may be false or badly argued. But why the hyperbolic words “clearly fallacious” and “misleading”? Fallacious and misleading to whom? To Dennett and to the other people who have exactly the same position on this as he does? It’s odd, then, that the tens of thousands of words which have been written on this subject were all written in response to an argument that is, apparently, clearly fallacious and misleading.

What’s worse, Dennett even hints (or perhaps explicitly states) that the Chinese Room argument is an argument specifically designed to please crowds! Of course Dennett may be arguing that crowd-pleasing has been an unintentional result of the argument. That said, judging from the rhetoric, hyperbole and amateur psychiatry Dennett indulges in (both here and elsewhere) -— I simply doubt that.

(3) Dennett: “You don’t want me to disable this device [this person’s “intuition pump”]; you like the conclusion so much — Strong AI is impossible, whew! — that your eyes glaze over at the prospect of being dragged through a meticulous critique of a vivid, entertaining argument that supports your fervent hope…. The details don’t really interest you, only the conclusion. What an anti-intellectual copout!”

Response: This is Dennett reading other people’s minds again!

What’s more, Dennett even appears to indulge in amateur psychiatry (or simply psychology) with his talk of “eyes glazing over, “fervent hope”, etc. He even accuses the people who hold different views on this of being (if not in these precise words) dunderheads, philistines and even plain dishonest.

Factually, I doubt that more than one in ten of the people who’re sceptical (or simply critical) of some of the claims of Strong A.I. would ever say that “Strong AI is impossible”. There’s no need for this modal hyperbole from Dennett. And, from what I’ve read, John Searle (1932-) himself has never made such an absolute claim about Strong AI.

(4) Dennett: “To many people consciousness is ‘real magic’. If you’re not talking about something that is supercalifragilisticexpialidocious, then you’re not talking about consciousness, the Mystery Beyond All Understanding.”

Response: As with all the other quotes from Dennett, admittedly there’s an element of truth in the words above. But only an element! In other words, it depends on which philosopher of consciousness he’s talking about and what exactly that philosopher argues. In addition, depending on the quotes in this selection, that degree of truth depends on precisely what Dennett claims about other people and their specific arguments regarding Strong AI.

So sure - there are some philosophers and many laypeople who see consciousness as once they saw God, the soul, the paranormal, ley lines, astral travelling, the flat earth, etc. Yet there are also many philosophers and laypeople who don’t have this kind of mindset on the supposedly supercalifragilisticexpialidocious nature of consciousness. Alternatively, even if they do, they may still not be woo merchants. In addition, there are those who believe that consciousness is a “Mystery”. However, they don’t also believe that it’s “Beyond All Understanding”. That is, they may simply believe that consciousness is a mystery at this present moment in time. So such people don’t believe that this must remain so for evermore.

(5) Dennett: “I am suggesting, then, that David Chalmers has — unintentionally — perpetrated the same feat of conceptual sleight of hand in declaring that he has discovered ‘The Hard Problem’.”

Response: This last quote from Dennett may be a little unfair because he does, after all, use the word “unintentionally” about David Chalmers’ position. However, I’m struggling to see how Chalmers could carry out a “conceptual sleight of hand” and do so “unintentionally”.

The ironic thing is that Chalmers is very sympathetic to artificial intelligence — even if only to Weak AI. (See Chalmers’ discussion with Dennett here.) It’s just that alongside his embrace of (weak) A.I. Chalmers does still believe that there’s a Hard Problem of consciousness.

And then there’s the gross sarcasm (or do I mean irony?) from Dennett again.

It seems that Dennett simply can’t accept that Chalmers has “discovered” the Hard Problem as a result of thinking deeply about the subject over many years. Instead, Dennett believes that Chalmers has “perpetrated” a “slight of hand”.

Conclusion

Having quoted all the above, the very mentioning of someone else’s ad hominems (rather than his arguments) could itself be deemed to be an example of an ad hominem! Then again, one shouldn’t take a pure (or absolute) position on ad homs. Sometimes they may be perfectly acceptable in philosophical writing — though only if they’re backed up by argument, data, etc. In any case, I said that I was surprised by Dennett’s ad homs. I also said they were crude. I didn’t say that ad homs — in and of themselves — are automatically a bad thing.

I’ll put more meat on that claim with my very own ad hominem.

For a long time I’ve believed that Dennett thinks that many — or even all — the philosophers who don’t agree with his philosophical views on this subject are… well, religious. Or, at the very least, Dennett believes that they have (not Dennett’s own words!) “secret religious leanings” which motivate their positions. Now that’s simply false. It may be true about some of Dennett’s critics. However, it’s certainly not true of all of them. And even if some of Dennett’s critics are indeed religious, he (as a philosopher) shouldn’t simply assume that they are. What’s more, he must still concentrate on their arguments.

Talking about religion — here’s my second ad hominem.

I certainly suspect that Dennett is a little dogmatic and even theological when it — specifically — comes to his behaviourist and verificationist positions on philosophical matters. So it can certainly be said that materialists (or physicalists/reductionists/verificationists/scientists/etc.) can be dogmatic — as can those who uphold literally any position on any subject. (Anti-materialists, for example, can be very dogmatic too.) This is why it’s wise to make a distinction between materialism (or reductionism/verificationism/etc.) and those people who uphold this philosophical position.

This means that materialism (or reductionism/etc.) itself can’t really be dogmatic partly because there are so many varieties of such a theory (or position). And surely an abstract philosophical position (theory) can’t be dogmatic in itself — only its human adherents can be.

[I can be found on Twitter here.]

A Short Note on Science and Empiricism

Not only is there empiricism within philosophy, there’s also an empiricist position towards science. Indeed some philosophers have argued that science itself is empiricist (at least in the past).

The Scottish philosopher Dave Hume (1711–1776) put this case very simply when, according to the Irish philosopher Ernan McMullin (in his 1984 paper ‘A Case for Scientific Realism’), he “restricted science to the patterning of sense impressions”. Of course this also — at least partly — stemmed from Hume’s well-known position on causality. And causality was nearly always seen (i.e., up until the late 19th and early 20th centuries) as the “cement of the universe” (J. L. Mackie). Indeed the rejection of causation (or, more correctly, necessary causal relations) was at the heart of empiricist philosophy. In Hume’s case (at least according to McMullin), he

“simply rejects the notion of cause according to which one could try to infer from these impressions to the unobserved entities causing them”.

There are obviously many problems with empiricism. More specifically, there are problems with “empiricist science”.

Take elementary particles.

The fact is that no one has ever observed an elementary particle (such as an electron or certainly a quark). However, people do observe things which lead them to believe that electrons exist.

Take the cloud chambers which are (or were) used by scientists to discover elementary particles and their nature. Charged entities (such as electrons) leave ionized tracks which betray their presence (or at least their former presence). Nonetheless, you still can’t say that you’ve observed an electron. All you can say is that you’ve observed an ionized track in a cloud chamber.

Alternatively, Ernan McMullin writes:

“An electron may be defined as the entity that is causally responsible for, amongst other things, certain kinds of cloud tracks.”

McMullin goes into more detail when he argues that an electron “will be said to exist [] if a number of convergent sorts of causal lines lead to it”.

There are many other simple reasons as to why an empiricist approach to science fails. Or the least you can argue is that empiricism is inadequate.

Take the Devonian geological period.

McMullin calls this period (or its postulation): “a theoretical entity”. It’s a theoretical entity primarily because it can’t be observed. However, clearly that doesn’t mean that we should reject it as a theory or even as a genuine period in the Earth’s history. In other words, even though the Devonian period can’t be observed, we can still say that this period existed roughly 400 to 350 million years ago. McMullin also says that during the Devonian period

“the dominant life form on earth was fish and a number of important developments in the vertebrate line occurred”.

The American theoretical physicist, string theorist and mathematician physicist Brian Greene (1963-) also offers some choice examples from the history of science.

Firstly, James Clerk Maxwell’s electromagnetic fields. Greene writes:

“James Clerk Maxwell’s architecture introduced a significant step in abstraction. Vibrating electric and magnetic fields are not the kinds of things for which our senses have evolved a direct affinity. Although we can see ‘light’ — electromagnetic undulations whose wavelengths lie in the range our eyes can detect — our visual experiences don’t directly trace the undulating fields the theory posits.”

Then elsewhere Greene mentions general relativity and quantum mechanics:

“Now, I’ve seen watches tick and I’ve used rulers to measure, yet I’ve never grasped spacetime in the same way I grasp the arms of my chair. I feel the effects of gravity, but if you pressed me on whether I can directly affirm that I’m immersed in curved spacetime, I find myself back in the Maxwellian situation… Probability waves give rise to predictions for where there this or that particle is likely to be found, but the waves themselves slither outside the arena of everyday reality.”

Shockingly (or perhaps not), then, it can be argued that this line of reasoning may lead us to happily embrace such things as the multiverse, strings, branes and whatnot, as it does in the particular case of Brian Greene. Whether these theories (or things) clash violently with empiricist science is, of course, an issue all on its own.

[I can be found on Twitter here.]

The Quantum World = The Mathematics

 First things first.

This essay may appear to advance two mutually-contradictory positions. On the one hand, it argues that without the mathematics (or, more correctly, without the mathematical formalism/s), there would be no quantum world — or at least no quantum mechanics. Yet, on the other hand, this essay also argues against Pythagoreanism — at least as it applies to this specific issue.

The main anti-Pythagorean argument in the following is that the world (or Nature) isn’t literally mathematics (whatever that may mean) or “made up” of numbers. It’s simply that, in quantum mechanics at the very least, without the mathematics, we’d have (almost) nothing.

Pythagoreanism: Things are Numbers

Pythagoreans believe that the world literally is mathematical. Or, perhaps more accurately, they believe that the world literally is (without the suffix “cal”) mathematics. I make this either/or distinction because if a physicist argues that “the world is mathematical”, then that may only mean that the world can be accurately — even very accurately — described by mathematics. The Pythagorean, however, states that “things are numbers”. Such a person therefore establishes a literal identity between the maths and the world (or parts thereof).

Yet we don’t need to accept the latter.

Having said all the above, it may well be the case that this essay does indeed advance a Pythagorean position — at least when it comes to quantum mechanics. That’s because it’s sometimes hard to tell what the Pythagorean position actually is. For one, it’s hard to make sense of the locution that the world is mathematical or that it’s made of numbers.

So the question which must now be asked is this:

What is it for “things” to be “numbers”?

To repeat: the Pythagorean position isn’t to only to argue that mathematics can describe (or model) things — it’s to argue that things literally are numbers. But what does that actually mean? And, as a consequence of that, it can now be asked if the statement “All things are numbers” is to be taken poetically or literally. Taken literally, it hardly makes sense. Taken poetically, it still requires much interpretation.

One interpretation of the Pythagorean position is that if things are numbers, then it’s no surprise that — for example — string theory is on top of things when it comes to describing reality. What I mean by that is this:

i) If things are numbers,
ii) and numbers are also used to describe (or model) things (which are numbers),
iii) then numbers are describing (or modelling) numbers.

That would mean that we never escape from numbers. Who knows, perhaps that’s precisely the result which Pythagoreans want!

To change tack a little.

The physicist John Archibald Wheeler (1911 — 2008) provided the best riposte to Pythagoreanism in physics. (I’m not entirely sure if this was his intention.)

It’s often been said that Wheeler used to write many arcane equations on the blackboard and stand back and say to his students:

“Now I’ll clap my hands and a universe will spring into existence.”

According to Pythagoreans, however, the equations are the universe.

And, after that comment, Steven Hawking (1942–2018) trumped Wheeler with an even better-known quote. He wrote:

“Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?”

The American science writer Kitty Ferguson (1941-) offered a (possible) Pythagorean answer to Hawking’s question. She suggested the possibility that “it might be that the equations are the fire”. Alternatively, could Hawking himself have been “suggesting that the laws have a life or creative force of their own?”. Again, is it that the equations are the fire?

So what, exactly, does “breathe[] fire into the equation [to] make a world”?

John D. Barrow

Mathematics is an extremely useful tool. The English cosmologist, theoretical physicist and mathematician John D. Barrow (1952–2020), however, went one step beyond that truism. Barrow actually put his — arguably — Pythagorean position in the following way:

“By translating the actual into the numerical we have found the secret to the structure and workings of the Universe.”

Of course almost everyone can happily accept the Universe and its parts are assigned numbers… Or are described by numbers… Or are captured by numbers… Or are explained by numbers… Or are (to use Barrow’s own words) translated into numbers. The thing is, that’s not actually a Pythagorean position.

So, as a consequence of all the words above, it’s no wonder that so many people have believed that through maths (as Barrow puts it) “we have found the secret to the structure and workings of the Universe”. Yet even here there must be a non-Pythagorean (as it were) remainder. What I mean by this is that maths finds the secret of things which already and separately exist — in this case, the “structure and workings” of the world. Surely it doesn’t also need to be argued that these structures and workings are literally mathematics (or literally numbers).

… Or perhaps it does.

To repeat: to the Pythagorean, the world and its parts are actually mathematical. This means that it isn’t that maths is simply helpful for describing the world — the world itself is mathematical. Indeed one must take this literally. Here’s Barrow again on the Pythagorean position:

“[The Pythagoreans] maintained ‘that things themselves are numbers’ and these numbers were the most basic constituents of reality.”

Barrow then became ever clearer when he continued in the following manner:

“What is peculiar about this view is that it regards numbers as being an immanent property of things; that is, number are ‘in’ things and cannot be separated or distinguished from them in any way.”

Moreover:

“It is not that objects merely posses certain properties which can be described by mathematical formulae. Everything, from the Universe as a whole, to each and every one of its parts, was number through and through.”

As stated earlier, it’s hard to grasp what the sentence “things themselves are numbers” even means. Can we really argue that reality and its parts are mathematics (as in the “is of identity”)? Can we really argue that reality and its parts are literally made up of numbers or equations? And can we even argue that reality and its parts somehow instantiate maths, numbers or equations?

Max Tegmark

The physicist and cosmologist Max Tegmark (1967-) also puts the contemporary case for Pythagoreanism in the following very concrete example:

“[If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure — indeed, that same mathematical structure.”

To spell out the passage above.

Max Tegmark isn’t simply arguing that mathematics is perfect for describing the “electricity-field strength” in a particular “physical space”. He’s arguing that the electricity-field strength is a “mathematical structure”. That is, the mathematics we use to describe the electricity field is one and the same thing as the electricity field. Thus, if that’s really the case, then the so-called “miracle of mathematics” is hardly a surprise! And that’s because — as already stated above — we essentially have a situation in which maths is describing maths. And if maths is describing maths, then the word “describing” is surely not the right word to use in the first place.

Tegmark gives us more detail on his position when he tells us that

“there’s a bunch of numbers at each point in spacetime is quite deep, and I think it’s telling us something not merely about our description of reality, but about reality itself”.

It can be argued that Tegmark contradicts himself in the above.

At one point Tegmark argues that a field “is just [ ] something represented by numbers at each point in spacetime”. Note here that we have the two words “something [my italics] represented”. Yet elsewhere Tegmark also argues that the field “is just” (or just is) a mathematical structure — the latter two words implying that all we have is number. To repeat: Tegmark argues that the field is “represented” by “three numbers at each point in spacetime”. Yet he doesn’t (in this passage at least) also say that the field is a set of numbers (or even a “structure” which includes numbers).

So perhaps there’s a difference between arguing that (as the original Pythagoreans did) “things themselves are numbers” and arguing that the world is mathematical. (I may be drowning in a sea of grammar here.) The latter may simply state that the world exhibits features which are best expressed (or described) by mathematics. The former, on the other hand, states that the world literally is mathematics.

Now take the case of string theory.

Michio Kaku and String Theory

Not only is string theory seemingly more dependent on mathematics than all the other areas of physics (though, of course, that can be debated), it seems that some physicists even see string theory as being a “branch of pure mathematics”.

The string theorist Michio Kaku (1947-), for example, doesn’t hide from this when he quotes a “Harvard physicist” saying as much. In Kaku’s own words:

“One Harvard physicist has sneered that string theory is not really a branch of physics at all, but actually a branch of pure mathematics, or philosophy, if not religion.”

After Kaku puts the Pythagorean position, he then quotes Albert Einstein (i.e., as backup) stating the following:

“‘I am convinced that we can discover by means of purely mathematical construction the concepts and the laws… which furnish the key to the understanding of natural phenomena.’”

Einstein went deeper when he added these words:

“‘Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it.’”

Now all the words above do indeed sound Pythagorean (or, more broadly, Rationalist) — at least on the surface. And Einstein seems to more or less come clean about this in his final sentence. Thus:

“‘In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.’”

The strange thing is that Kaku also seems to offer us a mutually-contradictory account (in his book Beyond Einstein) of Einstein’s position on mathematical physics. Kaku writes the following words:

“Einstein revealed a clue to the way he arrived at his great discoveries: he thought in physical pictures. The mathematics, no matter how abstract or complex, always came later, mainly as a tool by which to translate these physical pictures into a precise language.”

Indeed elsewhere in the same book Kaku also writes:

“[] Feynman, and other great scientists []thinks in terms of pictures that express the essential physical concept. The math comes later.”

If we return to Kaku’s own position.

As for the charge (if it is an charge) of Pythagoreanism against Kaku, don’t take my word for it: take the words of the man himself. Firstly Kaku lays out the essential Pythagorean position in this way:

“Not surprisingly, the Pythagoreans’ motto was ‘All things are numbers.’ Originally, they were so pleased with this result that they dared to apply these laws of harmony to the entire universe.”

Then Kaku continues by arguing that “with string theory” what we have is “physicists [] going back to the Pythagorean dream”.

Quantum Mechanics

Philip Ball

The science writer Philip Ball (1962-) argues (in his book Beyond Weird: Why Everything You Thought You Knew about Quantum Physics Is Different) that the mathematics of quantum mechanics “doesn’t say anything about the ‘real world’”. Many physicists — and some philosophers — have also echoed that sentiment. Yet that may appear to be an odd position. It’s odd because if the mathematics of quantum mechanics is extraordinarily successful when it comes to predictions, applications, engineering, technology and whatnot, then (perhaps almost by definition) surely it simply must be about the real world…

Yet what work is the word “real” actually doing here? Does it imply that we (or the maths) must mirror the world? But how does — or would — that work? And even if the maths perfectly describes physical phenomena in terms of their magnitudes, values, strengths/charges, velocities, spatial dimensions/positions, etc., then is all that actually a case of mirroring the world itself? Surely if the maths of quantum mechanics mirrored the world, then it would look — and even be — the same as that world. In that case, what purpose would such mirroring actually serve? (Think here of the often-made claim: The best model of x is x itself.)

So it can be argued that maths can’t — literally — mirror the world.

Despite saying that, one thing is still certainly the case.

As stated in the introduction, without the maths, we’d have almost (or even literally) nothing to say about the quantum world — real or otherwise. When it comes to the quantum world, the usual (as it were) means of ownership aren’t available to us. That is, we can’t observe, feel, smell or (often) even imagine the quantum world. Thus the maths is all we’ve got.

All this is excellently expressed in the following passage from the science writer John Horgan (1953-):

“[M]athematics helps physicists definite what is otherwise undefinable. A quark is a purely mathematical construct. It has no meaning apart from its mathematical definition. The properties of quarks — charm, colour, strangeness — are mathematical properties that have no analogue in the macroscopic world we inhabit.”

Thus if maths is all we’ve got, then it’s not really a surprise that many physicists (i.e., the more philosophical ones) argue that quantum mechanics doesn’t really say anything about the real world. (This has been said since Niels Bohr in the 1920s.) Or, at the very least, everything important — or even relevant — that’s said about the quantum world is said by the maths.

So when Philip Ball also writes that Richard Feynman could only do “quantum theory” (i.e., the maths), then that’s not a surprise. That’s because it can be argued that the maths is all we’ve got and all Feynman had. Indeed when we stray beyond the maths into interpretation, then we (perhaps by definition) can’t help but get things wrong.

Or at least that’s one (sceptical) scenario we must consider.

Again, it’s not a surprise that — even — Feynman didn’t “know what the maths means”. That may be because the words what the maths means are — almost — meaningless. At the very least, there’s a hint here that we can’t go beyond the maths. Yet it’s still the case that so many philosophers, and a somewhat lesser number of physicists, believe that the maths is only second best to something far… deeper.

John Gribbin

The British science writer and astrophysicist John Gribbin (1946-) appears to agree with these conclusions. That is, as a consequence of much of what’s been said above, it can be concluded that all the imagery, picture painting, metaphors, analogies, etc. we find in the popular accounts — and even the technical interpretations — of the quantum world are simply (to use Gribbin’s own words)

“crutches to help us imagine what is going on at the quantum level and to make testable predictions”.

Indeed Gribbin also believes that

“none of [the quantum mechanical interpretations] is anything other than a conceptual model designed to help our understanding of quantum phenomena”.

Indeed Gribbin also talks about the interpretations of quantum mechanics:

“I stress, again, that all such interpretations are myths, They are not, any of them, uniquely ‘the truth’; rather, they are all ‘real’, even where they disagree with one another.”

Many will read Gribbin’s words as being very radical — and even deflationary — when it comes to quantum mechanics. Yet, despite all the above, Gribbin also happily acknowledges that all these quantum interpreters genuinely believe that their very own interpretations are true. He writes:

“[T]he interpreters and their followers will each tell you that their own favoured interpretation is the one true faith, and all those who follow other faiths are heretics.”

And that passage comes straight after Gribbin had told us that

“[a]t the level of equations, none of these interpretations is better than any other”.

Thus, logically, “none of the interpretations is worse than any of the others, mathematically speaking”. That said, all this hinges on precisely how we’re supposed to take the phrases “at the level of equations” and “mathematically speaking”.

Gribbin also becomes very psychological (or aesthetic) when he concludes (as the very end of one of his books) that we are

“free to choose whichever one gives you most comfort, and ignore the rest”.

Again, there may well be an argument that all the interpretations of quantum mechanics are superfluous when it comes to predictions, tests, experiments, technology, etc. However, that certainly doesn’t mean that all these interpretations are “equally good”. They may all be equally bad in the sense that they don’t make the slightest bit of difference when it comes to to mathematical theory, predictions, (quantum) technology, etc. However, are they all equally good in literally every other respect?

Despite all Gribbin’s words above, he still stresses the importance 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 and on a final quasi-Pythagorean reading, it’s still clear that a “mathematical concept” comes first and only then is a physical model found to square with it.

[I can be found on Twitter here.]