Quantum Theory as Medieval Necromancy

The physicist Edwin Thompson Jaynes (1922–1998) once stated the following: “Somewhere in quantum theory, the distinction between reality and our knowledge of reality has become lost, and the result has more the character of medieval necromancy than of science.” This passage has been quoted many times. So what does it mean?

(i) Introduction

(ii) Quantum Theory and the Copenhagen Interpretation

(iii) Against Edwin Jaynes

(iv) Quantum Theory and Reality

(v) Peter van Inwagen’s “Ultimate Reality”

The words above are often quoted (see various citations here) in the critical context of quantum mechanics and its lack of commitment to reality.

This passage — and the rest of the paragraph it comes from — is, of course, hyperbolic. (See the complete paragraph at the end of this essay.) However, it can be excused (one can suppose) because it occurs at the end of an otherwise highly-technical paper… with lots of maths in it. (The paper is called ‘Quantum Beats’, which was published in 1980.)

So this passage is quoted again in this essay because it perfectly captures various (for want of a better word) realist positions on quantum mechanics.

Quantum Theory and the Copenhagen Interpretation

Edwin Jaynes’s critical account of quantum theory (or, more accurately, the Copenhagen interpretation) is broadly correct (i.e., once the rhetoric is cut out). Or, more accurately, it’s broadly standard.

For example, mathematical physicist Roger Penrose also writes:

“It is a common view among many of today’s physicists that quantum mechanics provides us with NO picture of ‘reality’ at all! The formalism of quantum mechanics, on this view, is to be taken as just that: a mathematical formalism. This formalism, as many quantum physicists would argue, tells us essentially nothing about an actual QUANTUM REALITY of the world, but merely allows us to compute probabilities for alternative realities that might occur. Such quantum physicists’ ontology — to the extent that they would be worried by matters of ‘ontology’ at all — would be the view (a): that there is simply no reality expressed in the quantum formalism.”

Then Penrose continued:

“At the other extreme, there are many quantum physicists who take the (seemingly) diametrically opposite view (b): that the unitarily evolving quantum state completely describes actual reality, with the alarming implication that practically all quantum alternatives must always continue to coexist (in superposition).”

So readers may now wonder if the alternative which Penrose explained above (perhaps also a hint at the many-worlds interpretation) was something that Jaynes himself endorsed. That is, did Jaynes accept Penrose’s “other extreme”? Well, not really.

Specifically, Jaynes used his mind projection fallacy against Penrose’s other extreme. In Jaynes’s own words:

“[I]n studying probability theory, it was vaguely troubling to see reference to ‘gaussian random variables’, or ‘stochastic processes’, or ‘stationary time series’, or ‘disorder’, as if the property of being gaussian, random, stochastic, stationary, or disorderly is a real property, like the property of possessing mass or length, existing in Nature. Indeed, some seek to develop statistical tests to determine the presence of these properties in their data [].

“Once one has grasped the idea, one sees the Mind Projection Fallacy everywhere. [] The error occurs in two complementary forms, which we might indicate thus: (A) (My own imagination) → (Real property of Nature), [or] (B) (My own ignorance) → (Nature is indeterminate).”

On Jaynes’s view, then, “gaussian random variables”, “stochastic processes”, “stationary time series” and (more broadly) “disorder” aren’t “real properties”. Instead, they’re basically stand-ins for real properties.

However, did the Copenhagenists — or any others physicists — ever take them to be real properties in the way that Jaynes portrays them? Instead, didn’t the Copenhagenists, etc. simply take them as functions, tools, devices, etc?

[Interestingly, critics of some of the claims found in information theory have said that information is often seen as — quoting Jaynes again, though on quantum theorists — “a real property, like the property of possessing mass or length, existing in Nature”.]

All that said, can’t it now be argued that the Copenhagenists were indeed realists about mathematical entities, rather than realists about physical properties?

[It’s worth noting here how Jaynes attempted to provide experimental and technical solutions to what others may see as the purely philosophical problems of interpretation. Jaynes, particularly, had a problem with such important ideas as the uncertainty principle and the divergences arising in quantum electrodynamics (see here), which I’m not really qualified to comment upon.]

To sum up. In the opening quote and elsewhere, Jaynes was essentially arguing against the Copenhagen interpretation of quantum mechanics. (Jaynes used the words “quantum theory”, rather than “quantum mechanics”.) Indeed, Jaynes fully endorsed the widely-held view that this interpretation is (somehow) defeatist, idealist or even mysterious.

So now let me put the opposite case — i.e., to Jaynes’s — equally rhetorically.

Against Edwin Jaynes

Edwin Jaynes’s position is — essentially — that if a scientific theory (or statement) doesn’t categorically offer us the (absolute?) truth, then it’s equivalent to “medieval necromancy”. Or, less rhetorically, Jaynes believed (just like Albert Einstein before him) that such (absolute) truth about the “real physical situation” (or “reality”) must be the primary aim of all physicists.

The essentials of Jaynes’s position would have been classed as “dogmatic realism” by Werner Heisenberg, who actually aimed that term at none other than Albert Einstein.

So now let the physicist and writer Paul Davies (1946-) sum up this battle between Jaynes’s (possible) realism and Copenhagenist (as it were) irrealism in the following paragraph:

“Einstein’s opinions are labelled ‘dogmatic realism’, a very natural attitude, according to Heisenberg. Indeed, the vast majority of scientists subscribe to it. They believe that their investigations actually refer to something real ‘out there’ in the physical world and that the lawful physical universe is not just the invention of scientists. The unexpected success of simple mathematical laws in physics bolsters the belief that science is tapping into an already existing external reality. But, Heisenberg reminds us, quantum mechanics is also founded on simple mathematical laws that are very successful in explaining the physical world but still do not require that world to have independent existence in the sense of dogmatic realism. So natural science is actually possible without the basis of dogmatic realism.”

Thus, in physics at least, Jaynes believed that there’s simply no room for modesty or the acceptance of any philosophical (or epistemic) limitations to the (as it were) realist aim of physics.

Thus, it’s not surprising that Jaynes stepped up his rhetoric when he claimed that such a defeatist irrealism “constitutes a violent irrationality”.

Yet surely it’s irrational to claim something (or claim anything) about “reality” which simply cannot be either known or experimentally justified. It’s irrational because it allows almost anything to be said about this… reality. Indeed, almost everything has been said about it. Hence the 15 (!) main (sometimes mutually-contradictory ) interpretations of quantum mechanics — plus the dozens of peripheral interpretations.

So this is a competitive war of rival interpretations of quantum mechanics (or Jaynes’s quantum theory) in which every interpretational warrior tells us what reality is and all his rivals either reject or deny his claims.

(The astrophysicist and writer John Gribbin put it this way: “[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.”)

All that said, was Jaynes only talking about the interpretations of quantum mechanics?

Quantum Theory and Reality

It’s interesting to note that Edwin Jaynes didn’t criticise the interpretations of quantum theory (or quantum mechanics): he criticised quantum theory itself.

For example, Jaynes told us that

“it is pretty clear why present quantum theory not only does not use — it does not even dare to mention — the notion of a ‘real physical situation’”.

Jaynes had a problem with this lack of (to use a philosophical term which has been said to be Einstein’s stance) realism.

[Some commentators dispute Einstein’s realism. So it often depends on which period of his life and which aspect of his work is being referred to. For example, this is the Stanford Encyclopedia of Philosophy categorically saying that Einstein was not a realist.)

So, tell me something about this “real physical situation” or (to use Jaynes’s other word) “reality” which doesn’t also refer to experiments, observations, theories, tests, etc. What is it like?

That said, it can be supposed that it’s indeed (strictly) true that quantum theory doesn’t “use” the “notion of reality”. It certainly doesn’t “mention” such a thing. That’s because these words are either reifications or even anthropomorphisms regarding quantum theory.

So can an almost purely-mathematical theory “mention” reality? In what way would that work? Is a mathematical theory (or the quantum formalism) really meant to do that job?

On the other hand, many physicists, as interpreters of quantum theory, have both used and mentioned reality.

Now let’s take a detour in which it can be argued that Edwin Jaynes’s position on physics is similar to Peter van Inwagen’s position on metaphysics.

Peter van Inwagen’s “Ultimate Reality”

The American philosopher Peter van Inwagen (1942-) offers his readers some rhetorical accusations which are very much like Edwin Jaynes’s.

For example, van Inwagen uses the words Orwellianism, “relativism” and “idealism” against views which aren’t (his own?) metaphysical realism.

So take the final paragraph of Peter van Inwagen’s chapter ‘Objectivity’. In this chapter, van Inwagen also attacks what he calls “anti-realism” and mentions George Orwell’s novel Nineteen Eighty-Four in the process. He writes:

“[] I should like to direct the reader’s attention to the greatest of all attacks on anti-Realism, George Orwell’s novel 1984. Anyone who is interested in Realism and anti-Realism should be steeped in the message of this book. The reader is particularly directed to the debate between the Realist Winston Smith and the anti-Realist O’Brien that is the climax of the novel. In the end, there is only one question that can be addressed to the anti-Realist: How does your position differ from O’Brien’s?”

[See also van Inwagen’s ‘Was George Orwell a Metaphysical Realist?’.]

Van Inwagen seems to see anti-realism as some kind of postmodernist fashion designed to let anything go. Or, perhaps, van Inwagen sees anti-realism as advancing various “radical” political projects.

As it is, George Orwell’s Winston Smith isn’t really a realist at all. Or, rather, he’s neither a realist nor an anti-realist. That’s primarily because the dispute between anti-realism and realism (as least as expressed and characterised by van Inwagen) is largely a late-20th-century phenomenon within the domain of Anglo-American analytic philosophy; as well as, to a much lesser degree, within physics and some of the other sciences.

Interestingly enough, the cognitive scientist and linguist Steven Pinker almost replicates — as well as giving more detail — van Inwagen’s political positions when he himself mentions Orwell’s O’Brien. (This time in relation to postmodernism, not anti-realism.)

Firstly, Pinker quotes directly from Nineteen Eighty-Four:

“‘You believe that reality is something objective, external, existing in its own right. [] But I tell you, Winston, that reality is not external. Reality exists in the human mind, and nowhere else. Not in the individual mind, which can make mistakes, and in any case soon perishes; only in the mind of the Party, which is collective and immortal.’”

Van Inwagen also sees idealism as being indistinguishable from anti-realism when it comes to what really matters… philosophically. Indeed, the philosopher Michael J. Loux — van Inwagen’s fellow University of Notre Dame metaphysical realist — sees “subjective idealism” as “the view that we make it all up”.

So whereas Edwin Jaynes demanded (some kind of) realism from physicists, van Inwagen demands metaphysical realism from philosophers.

Van Inwagen also tells us that

“metaphysics is the attempt to discover the nature of ultimate reality”.

Thus, if a philosopher doesn’t accept van Inwagen’s definition (or if a philosopher questions the poetic term “ultimate reality”), then he can’t be doing metaphysics at all! Similarly, Jaynes believed that physicists are indulging in “mediaeval necromancy” and “irrationality” if they didn’t accept his own philosophical position on quantum theory.

So it’s not a surprise that van Inwagen also says the following:

“It is therefore misleading to think of anti-Realism as a metaphysics. [] Anti-Realism, rather, is a denial of the possibility of metaphysics [].”

It’s true that anti-realists emphasise the epistemological approach to metaphysics. And the Copenhagenists — and many other physicists - have also emphasised the epistemological approach to quantum mechanics.

Yet anti-realism is still an approach to metaphysics. It isn’t automatically a denial of the possibility — or existence — of metaphysics. However, it is if one accepts van Inwagen’s position in full. Indeed, it seems that one has to accept van Inwagen’s position in full if one wants to continue doing metaphysics.

To repeat: if one doesn’t abide by van Inwagen’s take on both realism and anti-realism (as well as his take on metaphysics itself), then, according to van Inwagen himself, one can’t be doing metaphysics at all.

It seems that something similar can also be said about Edwin Jaynes and his position on what he called “quantum theory”.

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Note:

“From this, it is pretty clear why present quantum theory not only does not use — it does not even dare to mention — the notion of a ‘real physical situation.’ Defenders of the theory say that this notion is philosophically naive, a throwback to outmoded ways of thinking, and that recognition of this constitutes deep new wisdom about the nature of human knowledge. I say that it constitutes a violent irrationality, that somewhere in this theory the distinction between reality and our knowledge of reality has become lost, and the result has more the character of medieval necromancy than of science.” — Edwin Jaynes

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David Chalmers’ Unanswerable Questions: “Why do I have THIS experience?” and “Why do we see red, rather than hear a trumpet?”

The philosopher David Chalmers believes that there are answers — or just possible answers — to his questions about conscious experiences. But what if there aren’t any answers — or even any possible answers? So perhaps it’s fair to say that Chalmers simply assumes that there are answers to his questions.

Left: W.H. Auden. Right: David Chalmers. The source of this quote can be found here.

Firstly, David Chalmers isn’t asking for answers which refer to anything physical, functional, structural, etc.:

(1) He isn’t asking for the “physical correlates” of experiences. 
(2) He isn’t asking for the causal and physical connections between the brain and experiences — or any connections between experiences and anything physical. 
(3) He isn’t asking for the functional (or otherwise) underpinnings of experiences.
(4) And he isn’t asking for any evolutionary answers either.

As Chalmers is keen to stress, all the physical things referred to above can be instantiated, and the experiences may still not occur. Alternatively, all these things can be instantiated and the experiences could still be very different.

[All David Chalmers’ questions in this essay come from his book, The Conscious Mind: In Search of a Fundamental Theory.]

Question (or Questions) 1

“At a more basic level, why is seeing red like THIS, rather than like THAT? It seems conceivable that when looking at red things, such as roses, one might have had the sort of color experience that one in fact has when looking at blue things. Why is the experience one way rather than the other? Why, for that matter, do we experience the reddish sensation that we do, rather than some entirely different kind of sensation, like the sound of a trumpet?”

[The source of this passage can be found here.]

Surely there are evolutionary and physical reasons as to why “seeing red is like THAT”. However, these reasons won’t satisfy Chalmers.

In other words, whatever reasons are given, Chalmers can still ask:

But why is experience E like THAT?

From an evolutionary perspective (perhaps also from a purely chemical and sensory perspective too), it wouldn’t make sense to hear a trumpet when looking at a red rose. Indeed, there’s an entire evolutionary history of sound and its relation to species’ sensory systems that can explain all this.

Yet Chalmers would say that none of that would explain the redness of a red rose or the trumpety sound of a trumpet when being blown. And it wouldn’t explain why we see red rather than hear the sound of a trumpet when we look at a (red) rose.

So exactly why is it conceivable that human beings could, say, smell dogshit when they look at a red rose?

What, exactly, is being conceived here?

More strongly, are these things actually being conceived in the first place?

After all, if I ask:

What, exactly, are you conceiving?

And Chalmers answers:

I’m conceiving that it’s possible to smell dogshit when I look at a red rose.

Then that’s neither an answer nor an explanation. Chalmers (or anyone else) would simply be uttering a single statement.

Chalmers will of course argue that he’s not talking about imagining smelling dogshit when he looks at a red rose. He’d claim that he’s simply conceived that this is possible. (Here’s one Cartesian account of this distinction.)

To repeat. Chalmers isn’t claiming to be carrying out any acts of imagination in which when he looks at a red rose he smells dogshit. He’s claiming that it’s conceivable that when he looks at a red rose, he could smell dogshit.

But isn’t that conceiving-imagining distinction a difference which doesn’t really make a difference?

[In the following, the panpsychist Philip Goff commits himself to this distinction: “The zombie argument is generally known in the academic philosophical literature as the ‘conceivability argument.’ I think this is something of a misnomer, as it suggests that the argument has something to do with what can be imagined.”]

Alternatively, Chalmers is also saying that he wants to know why we don’t smell dogshit (or see blue) when we look at a red rose.

Yet there may be evolutionary, structural, physical, etc. reasons for seeing red when looking at a (red) rose.

Now take this reformulation of Chalmers’ question:

Given that water is H₂O, why does it have this “particular nature”?

Chalmers would, of course, reject this parallel between H2O/water and physical states/experience.

Firstly, Chalmers would (correctly) say that water simply is H₂O. However, he’d also add that experience E isn’t brain state B — or any physical conglomerate. (Chalmers believes that E isn’t anything physical at all.) In other words, experience E is over and above any physical conglomerate C, whereas water isn’t over and above the molecule (or set of molecules) H₂O.

So the fact that Chalmers already believes that experience E isn’t physical means that he’s always free to ask his initial question. If E were physical, on the other hand, then his question would make less sense.

That said, even if experience E isn’t physical, then it may still the case that Chalmers’ question can’t be answered. It may be a brute fact.

[Chalmers, incidentally, accepts brute facts in other areas of science, philosophy and logic. So why not here? For example, Chalmers tells us that physics “does not tell us why there is [matter] in the first place”. So it may not be able to tell us why many properties in physics “have their nature”. Thus, such things are deemed to be — to use Chalmers’ own word — “primitive”. That is, they can’t be “deduced from more basic principles”.]

Of course we’d need to explain what brute facts actually are and why we should accept them. More relevantly, why should we accept that experience E’s nature is simply a brute fact?

So a (to use Valerie Gray Hardcastle’s term) “water-mysterian” can now ask:

Why does H₂O have its particular nature?

That is, he can also ask:

Why is H₂O wet and transparent?

Well, chemists, neuroscientists, physiologists, etc. can provide an answer to these questions.

Yet, as stated before, Chalmers wouldn’t accept the parallel between water and experience E. That is, chemists (along with neuroscientists, evolutionary theorists, physicists, etc.) can provide a physical, causal, structural, evolutionary, etc. story as to why water is wet and transparent. However, Chalmers will argue that no physical, causal, structural, biochemical, functional, etc. story will tell us why experience E has its particular nature (say, why it is red).

However, is Chalmers right to believe that the two cases aren’t at all parallel?

Now take this evolution- and sensory-based account of water’s transparency:

“Water is transparent because eyes first evolved in water. The range of the electromagnetic spectrum we detect corresponds to the spectrum for which water is transparent (absorbs the least). Had we evolved in mercury, we would think mercury is transparent and detect EM waves that pass through mercury.”

Yet this part-evolutionary account can still elicit a Chalmersesque question:

Yes, but why transparency?

So Chalmers can still ask his question after being supplied with a ton of such evolutionary and other physical (or chemical) reasons. After all, if there are evolutionary reasons as to why dogshit smells bad, then the question can still be asked as to why it smells bad or why it smells bad in that particular way.

Question 2

“When I open my eyes and look around my office, why do I have THIS sort of complex experience?”

Chalmers had to have some kind of “complex experience” when he looked around his office. And cognitive scientists, neuroscientists, etc. would happily answer his question — and do so in great detail. However, Chalmers will still ask:

“[But] why do I have THIS sort of complex experience?”

Again, the physical, causal, structural, etc. story about things in the world and their relations to experiences will always leave Chalmers unhappy because he can still ask his question.

Question 3

“Given that conscious experience exists, why do individual experiences have their particular nature?”

We have x and we have y. x and y are taken to be different things. This means that because y isn’t x, then we can always ask why x and y must be placed (or come) together or whether y really tells us everything about x.

However, what if x and y are one and the same thing under different modes of presentation? This would mean that any questions as to why x and y occur together, or why y is correlated with x, would make less sense.

Yet even if x = y, Chalmers could still ask his questions. That is, even if a conglomeration of physical conditions and particular experience were one and the same thing, then Chalmers could still ask why that experience is the way it is. So even if an identity were established, Chalmers could still ask his question.

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Note: The H₂O/water and physical/experience comparison will be tackled in more detail soon.

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Did Mathematics “Know” the Universe is Expanding, When Einstein Didn’t?

A New Scientist writer asks her readers two questions: (1) “How did Einstein’s equations ‘know’ that the universe was expanding, when he did not?” (2) “How is it possible that mathematics ‘knows’ about Higgs particles?”… What do these questions mean? Are they anthropomorphic in nature?

(i) Introduction: Mathematics Knows Things

(ii) Einstein Rejects the Universe’s Expansion

(iii) Input and Output

(iv) Pure Maths and Describing the World

(v) Eugene Wigner

(vi) Max Tegmark

(vii) Lee Smolin

Introduction: Mathematics Knows Things

In an article called ‘Reality: Is everything made of numbers?’, the New Scientist’s Amanda Gefter sets the scene in the following way:

“When Albert Einstein finally completed his general theory of relativity in 1916, he looked down at the equations and discovered an unexpected message: the universe is expanding.”

However:

“Einstein didn’t believe the physical universe could shrink or grow, so he ignored what the equations were telling him.”

What is directly relevant to this essay is Gefter’s following question:

“How did Einstein’s equations ‘know’ that the universe was expanding, when he did not?”

Interestingly, Amanda Gefter then applies the very same reasoning to Higgs particles. Indeed, she even uses the word “knows” again. (Which is also put in scare quotes.) She writes:

“How is it possible that mathematics ‘knows’ about Higgs particles or any other feature of physical reality?”

These questions have an anthropomorphic ring to them. Indeed, they’re more anthropomorphic (see here too) than some comments about ants or dolphins

So is Amanda Gefter excused from accusations of anthropomorphism simply because she puts the word “know” in scare quotes?

The problem here is that if her words aren’t taken literally, then it’s hard to think of an alternative way of taking them.

So it all depends.

Perhaps Gefter’s use of the word “know” is, at least partly, explained in the following passage from her article:

“‘Maybe it’s because math is reality,’ says physicist Brian Greene of Columbia University, New York. Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.”

This maths-is-reality stance will be tackled later.

So now let’s return to Gefter’s comments on Einstein ruling out an expanding universe.

Einstein Rejects the Universe’s Expansion

Much has been written about the scientific, philosophical and even religious reasons why Einstein might have (initially) ruled out the expansion of the universe. So these reasons may explain why he also rejected (to use a phrase used by many writers about many physicists) “what the mathematics was telling him” or what the maths knew.

However, the maths might not have been (as it were) running off in its own direction at all. Instead, Einstein might have simply rejected his own equations for all the reasons just mentioned. [See Einstein’s ‘Physical cosmology’.]

So it was still (perhaps paradoxically) Einstein’s own maths (or equations) which supposedly knew stuff which he didn’t know. That is, it wasn’t someone else’s maths. And it wasn’t (as it were) math’s very own maths either.

This means that maths itself (or maths alone) didn’t know that the universe is expanding.

That’s mainly because maths — i.e., on its own — doesn’t include the notions of the universe, expansion, gravity, space, matter, mass, etc. These are terms from physics and cosmology, not (pure) mathematics.

Thus, the equations which Einstein both created and used led to (physical) consequences which Einstein rejected. However, that didn’t mean that there was any genuine independence of the equations from Einstein himself. (This isn’t a reference to maths — as it were — in the abstract, but to the equations which Einstein himself created.) After all, if Einstein hadn’t recognised his famous cosmological “blunder”, then the maths still couldn’t have known anything he didn’t know. And, again, Einstein arguably rejected his own equations for reasons that had nothing to do with maths. Yet it was still his own equations which he rejected. That is, the equations which Einstein rejected didn’t create themselves, let alone show that they has applications to the notions in physics which were around in the early 20th century.

Input and Output

The New Scientist’s Amanda Gefter continues with this basic input-output scenario:

“If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it churn out anything beyond what we put in?”

We humans “put in” all sorts of stuff into all sorts of things. These things “churn out” all sorts of other stuff which is different to what we put in.

For example, we put coal into a stove and it churns out heat and smoke. We put data into a computer and it churns out all sorts of information which we didn’t put in.

There are literally innumerable examples of this.

So what we put in is transformed into something else. That something else is still a byproduct of what we put in. Thus heat and smoke are byproducts of putting coal in a stove. And, as many people who’re critical of the claims of artificial intelligence are keen to tell us, computers wouldn’t churn out anything if we hadn’t firstly put in the data (as well as if we hadn’t built the computer in the first place).

So perhaps this New Scientist writer has something distinct in mind when it comes to mathematics.

Well, mathematics is definitely distinct from a stove and what we we put into it. Similarly, its not like a computer or the data we put into it (though mathematical data can be fed into a computer).

But so what?

A stove isn’t a computer either. And an apple isn’t an orange.

Amanda Gefter also says that some (or many) scientists believe that maths is “nothing more than a language we use to describe the world”.

Pure Maths and Describing the World

Very few mathematicians and physicists have ever claimed that mathematics “is nothing more than a language we use to describe the world”. There is, after all, such a thing as pure mathematics (see also ‘Applied Mathematics’). That is, there is much maths which doesn’t — and perhaps even couldn’t — have any use in terms of “describing the world”.

This may be debatable, however.

Even some arcane mathematics in history came to have a use in physics. However, such maths obviously had a previous independence from physics for the simple fact that it existed for years — even hundreds of years — before physicists found a use for it.

Similarly, even those people who claim that maths is (to use Gefter’s words again) “an invention of the human brain” don’t see it simply in terms of its use in describing the world. So maths can be such an “invention”, and yet still have no use in physics — or anywhere else.

Predictably, Amanda Gefter then mentions and quotes the theoretical physicist Eugene Wigner (1902–1995).

Eugene Wigner

Gefter writes:

“‘It is difficult to avoid the impression that a miracle confronts us here,’ wrote physicist Eugene Wigner in his classic 1960 paper ‘The unreasonable effectiveness of mathematics in the natural sciences’ (Communications on Pure and Applied Mathematics, vol 13, p 1).”

Wigner himself also wrote:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

So, relevantly, now let’s requote Gefter quoting Brian Greene:

“‘Maybe it’s because math is reality,’ says physicist Brian Greene of Columbia University, New York. Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.”

Isn’t it best to state that reality is maths (or, less strongly, reality is mathematical), rather than Brian Greene’s “math is reality”? At least that’s how Pythagoreans and many physicists have put it over the years. That said, if you reverse a mathematical identity, then nothing is really changed. Thus if we have 2 + 2 = 4, and then reverse it to 4 = 2 + 2, then we get the same result. So perhaps stating that maths is reality is the same as stating that reality is maths.

In any case, why does it automatically follow that maths knows things simply because maths is reality?

That is, even if maths is reality, it would still require physicists to know that. Physicists also need to realise that maths and reality are one and the same thing. That is, if maths is reality (or if reality is maths), then physicists would still need to construct the equations and theories which help show us that that this is the case.

Indeed, if there is a necessary — and indeed blindingly obvious — contribution from physicists to this (as it were) maths = reality equation, then that equation may not hold at all. After all, physicists often get the maths-of-reality wrong. They also offer us contradictory maths-of-reality.

The physicist and cosmologist Max Tegmark also mentions Eugene Wigner a couple of times (i.e., in his book Our Mathematical Universe). Tegmark is clearly inspired by Wigner’s well-known questions and points.

Max Tegmark

So we have the following passage from Wigner, which Tegmark quotes:

“The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”

Albert Einstein also asked the same question in the following:

“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

… But hang on a minute!

Einstein’s following oft-quoted conclusion (as found in his ‘Geometry and Experience’) appears to be radically at odds with both Wigner’s and Tegmark’s positions:

“[] In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

So does the “unreasonable effectiveness” of electricity or roads also “demand an explanation”? That ironic question is asked because there are indeed explanations of maths effectiveness. However, I feel that they won’t satisfy mathematical mysterians like Max Tegmark.

The theoretical physicist Lee Smolin (1955-) also refers to “the obvious effectiveness of mathematics in physics”.

Lee Smolin

Smolin initially refers to the purely pragmatic utility of mathematics when it comes to physics.

Thus, we can all happily accept the unreasonable effectiveness of mathematics in physics.

But where do we go from there?

Smolin himself goes on to state that he has

“never heard a good a priori argument that the world must be organised according to mathematical principles”.

Again: mathematics is useful — extremely useful — in physics. So much so that there wouldn’t be any modern physics without maths. That said, it still can’t be concluded from this effectiveness that (to use Smolin’s words again) “the world must be organised according to mathematical principles”.

In other words, the unreasonable effectiveness of mathematics doesn’t mean — or have the consequence — that the world itself is (or must be) organised according to mathematical principles. Of course, it gives physicists reasons — even very good reasons — to believe that. However, the effectiveness of mathematics in physics — alone — doesn’t have the (logical) consequence that the world itself must be organised according to mathematical principles.

And it certainly doesn’t mean that (as Brian Greene put it) “math is reality” (or that reality is maths).

More specifically, when Smolin uses the words “a good a priori argument” (or simply when he uses the epistemological term a priori), he seems to be saying that many physicists simply assume that “the world” (or Nature) is mathematical precisely because of the unreasonable effectiveness of mathematics in physics…

All this is very close to being a circular position. Thus:

(i) Mathematics is unreasonably effective in physics because the world itself is mathematical. 
(ii) Because the world itself is mathematical, it logically follows that the mathematics in physics will be unreasonably effective.

However, isn’t the above like making the following (admittedly much weaker or less sexy) claim? —

(i) Cement is unreasonably effective when it comes to building houses. 
(ii) Therefore houses must be built on cement-based principles.

Later on in the same chapter, Smolin goes on to be even more explicit about these assumptions when he writes the following words:

“[W]hat is both wonderful and terrifying is that is absolutely no reason that nature at its deepest level must have anything to do with mathematics.”

At first sight, this seems like an incredible claim.

Or at least one would presume that many— or even most — physicists would have (deep?) problems with Smolin’s statement.

However, that shock may simply be down — again — to the false inference (which many physicists make) from the the unreasonable effectiveness of mathematics to the conclusion that nature itself (“at its deepest level”) must be mathematical.

To repeat: we have the following line of reasoning from some (or even many) mathematical and theoretical physicists:

(i) Mathematics is unreasonably effective in physics. 
(ii) Therefore the world itself must be organised according to mathematical principles.

Now it must be borne in mind that not all (or even most) physicists actually express (or even think — in great detail — about) these almost purely philosophical issues.

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