Thursday, 10 October 2019

Michio Kaku's Pythagorean & Postmodern String Theory

i) Old-style Physics
ii) Galileo
iii) Einstein
iv) Pythagoreanism
v) The Beauties and Symmetries of String Theory
vi) String Solutions and the Multiverse

[All unlinked quotes are from Michio Kaku's Parallel Worlds.]

Much has been made of string theory being “unscientific”, lacking evidence and not offering unique predictions. This piece lays part of the blame at the Pythagorean nature of string theory (in which the mathematics always has a supreme position).

The first section tackles what's called “old-style physics” - that is physics that places an importance on observation, experiment and prediction. Then there's a section in which Michio Kaku ties his own position to the science of Galileo and then Einstein. In both cases, it can be seen that they aren't great exemplars for string theory.

The central section deals with Pythagoreanism and how it ties in with string theory. The next section deals with the aesthetics of string theory and argues that aesthetic appeal can't be – and mustn't be - the end of the story in physics.

Finally, there's a section which cites a specific example of how string theory mathematics can lead to string theory physics and cosmology: i.e., the case of string solutions and the multiverse.


Michio Kaku is an American theoretical physicist and populariser of science. He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center.

Old-style Physics

The main contention of this piece is that string theory is essentially Pythagorean in nature. String theory has also been described as “postmodern” by the theoretical physicist Lee Smolin. In terms of the latter description, this doesn't mean postmodern in the philosophical sense practised by philosophers like Jean Baudrillard, Jean-François Lyotard, etc.: it simply means after modern physics. That is, this is how physics was done when string theory arrived. (Having said that, no doubt Smolin was aware of the philosophical associations of this term.) Smolin puts it this way:

The feeling was that there could be only one consistent theory that unified all of physics, and since string theory appeared to do that, it had to be right. No more reliance on experiment to check our theories. That was the stuff of Galileo. Mathematics [alone] now sufficed to explore the laws of nature. We had entered the period of postmodern physics.”

The writer Jim Holt puts the same point in this way:

For the first time in its history, theory has caught up with experiment. In the absence of new data, physicists must steer by something other than hard empirical evidence in their quest for a final theory.”

Kaku himself (though talking about cosmology in the 1960s) also uses words which are perfectly apt for string theory. He writes:

[Cosmology/string theory] was not an experimental science at all, where one can test hypotheses with precise instrument, but rather a collection of loose, highly speculative theories.”

Kaku also tells us how physics used to be before string theory:

In the past, physics was usually based on making painfully detailed observations of nature, formulating some partial hypothesis, carefully testing the idea against the data, and then tediously repeating the process, over and over again.”

Despite the above being a very simplified picture (which Kaku wouldn't deny), Kaku pits it against the approach employed by string theorists. Kaku continues:

String theory was a seat-of-your-pants method based on simply guessing the answer. Such breathtaking shortcuts were not supposed to be possible.”

Just as it was said that Kaku's account of “old-style” physics is simplified, so we can say that same about his account of the early days of string theory (despite him being an important member of that group). Nonetheless, despite these simplifications and exaggerations, Kaku and many others are well aware that string theory is seen as being both postmodern and Pythagorean. So it seems odd (at least prima facie) that a string theorist would admit that string theory was based on “simply guessing”. Then again, guessing (or at least speculation) has always been a part of physics. So string theory isn't unique in that respect. And even critics of string theory may have a problem with that emotive word – i.e., “guessing”.

Anyway, even according to a string theorist himself (i.e., Kaku), whereas old-style physics involved observations, tests and experiments; string theory is about maths... and guessing.

Not only is string theory seemingly more dependent on maths than other areas of physics (though that can be debated), it seems that some physicists even see string theory as being a branch of mathematics. Kaku doesn't hide from this because 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.”

That Harvard physicist was “Nobel Laureate Sheldon Glashow”. So can we now play Kaku's emotive word “sneered” against Glashow's equally emotive word “religion”?

We can see that although string theory has an hegemony in physics (at least according to Lee Smolin, Sheldon Glashow and other physicists), there are some physicists who are very unhappy with this. Kaku again puts the case of the Opposition in this respect. According to Kaku, Glashow “compar[ed] the superstring bandwagon to the Star Wars program (which consumes vast resources yet can never be tested”.

Glashow then got technical when he also said (at least according to Kaku) that

string theory will dominate physics the same way that Kaluza-Klein theory (which he considers 'kooky') dominated physics for the last fifty years, which is not at all.”

Of course talk of the Kaluza-Klein theory (“which posited a fifth dimension”) is relevant here because that was the theory which introduced an extra dimension into physics. That may mean that Glashow believes that the rot actually set in during the 1920s.


Kaku mentions Galileo in support of his Pythagorean position. (Kaku never uses the word 'Pythagorean' about his own position.)

Galileo was also at least partly Pythagorean in that he noted the vital importance of numbers in physics. Yet he was also fundamentally un-Pythagorean in the respect that he often moved from the world to the maths - rather than vice versa. That is, he both observed the world and carried out experiments; which the Pythagoreans never did. Thus it’s no surprise that Kaku quotes this well-known passage from Galileo:

'[The universe] cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word.'”

Galileo is basically saying that mathematics comes first. That is, “until we have learned the language and become familiar with the characters in which it is written”, we simply can’t understand the universe. However, that didn't stop Galileo from moving in both directions: from maths-to-world and then from world-to-maths. (The same is true of the comments from Einstein later.) This hints at the conclusion that the either maths-to-world or world-to-maths idea is, in fact, a false binary opposition. At least it's a false binary opposition when it comes to Galileo and, later, Einstein. However, it may be a true binary opposition when it comes to Pythagoreans, strings theorists and Max Tegmark (see later comments).

In addition, perhaps I'm doing Galileo a disservice here because he did say that

“we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”.

Yes, Galileo was talking about our understanding of Nature - not just Nature as it is (as it were) “in itself”. Nonetheless, Galileo also said that the “book is written in mathematical language”. So was he also arguing that Nature as it is in itself is mathematical? That is, perhaps Galileo wasn't only saying that mathematics is required to understand Nature.

There is, therefore, an ambivalence here between the following:

i) The idea that Nature itself is mathematical. And
ii) The idea that mathematics is required to understand Nature.

Surely we must now say that “Nature's book” isn't written in the language of mathematics. Sure, we can say that Nature's book can be written in the language mathematics. (Indeed it often is written in the language of mathematics.) Though Nature's book is not in itself mathematical because that book - in a strong sense - didn't even exist until human beings began to write (some of) it.


After Kaku puts the Pythagorean position, he then mentions Albert Einstein. He quotes Einstein (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 this does indeed sound rationalist or Pythagorean – at least on the surface. And Einstein more or less came 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.'”

Yet doesn't all this seem back to front? That is, Einstein claimed to be working from a “purely mathematical construction” and from there he went on to discover new “concepts and laws”. In other words, mathematical concepts “furnish[ed] the key to the understanding of natural phenomena”.

Nonetheless, it's not a (purely) rationalist position to claim that mathematical concepts can “furnish the key to the understanding of natural phenomena”. Basically, most (or even all) physicists would accept that. However, claiming that through “pure thought” alone we can “grasp reality” is surely a rationalist position. Then again, pure rationalists wouldn’t say that “experience may suggest the appropriate mathematical concepts”. And they would rarely talk of “natural phenomena”. So, in that sense, it’s clear that pure rationalism is a rare position. And it’s even rarer when it comes to physicists (i.e., rather than philosophers).

This means that in the Einstein passage above there's a subtle merging of rationalism and empiricism - in a Kantian mode. That Kantian mode is best expressed in this sentence from Einstein:

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

This suggests (in a true Kantian spirit) that the mathematical concepts are already there (as it were) in the human mind and only then do the observations of physicists help to tease them out. In any case, Kant certainly incorporated what Einstein called “mathematical concepts” into his “a priori reasoning”.

Does this off-piste journey into Einsteinian rationalism validate Kaku’s previously-quoted claim? (The claim that “verification of string theory might come entirely from pure mathematics, rather than experiment”?) Probably not. That's primarily because Kaku doesn’t elaborate on his philosophical position (as Einstein does). The other thing is that many of Einstein’s forays into philosophy aren't very sophisticated. That’s not a surprise since he wasn’t a philosopher. Thus the quote above may well contain contradictions or unclear reasoning. It’s also the case that Einstein often changed his philosophical positions throughout his life.

For example, Kaku again mentions Einstein (in the same book) stating this:

Einstein once said that if a theory did not offer a physical picture that even a child could understand, then it was probably useless.”

On the surface, Einstein’s position here is far from being rationalist or Pythagorean. Indeed even an hard-core empiricist probably wouldn’t feel the need to go as far as Einstein did (at least at that moment in his career). An empiricist may concede that “physical interpretations” of theories or mathematical concepts are important. However, we shouldn’t get too fixated on mental imagery or “physical picture[s] that even a child could understand”.

And here's another example which Kaku conveniently offers us. This time, however, it's Kaku talking about Einstein's position. Kaku writes:

To Einstein, any solution of his equations, if it began with a physically plausible starting point, should correspond to a physically possible object.”

This again shows Einstein in non-rationalist mode and perhaps it works as a warning to string theorists. It also squares perfectly well with the astrophysicist and writer John Gribbin on the same subject. Gribbin says 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”.

When it comes to Einstein's possible philosophical confusions, it's worthwhile noting that Kaku also quotes Einstein's warnings against philosophy. Einstein is quoted as stating the following:

“'Is not all of philosophy as if written in honey? It looks wonderful when one contemplates it, but when one looks again it is all gone. Only mush remains.'”

So perhaps Einstein should have heeded his own warning.


As for the “accusation” of Pythagoreanism (if it is an accusation), don’t take my word for it. Take the words of Michio Kaku himself. Firstly Kaku lays out the essential Pythagorean position:

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 saying that with string theory, physicists are going back to the Pythagorean dream”.

As will be expressed later in this piece, the intellectual move for the Pythagoreans (at least as Kaku expressed it) was from maths (or “numbers”) to the universe/world/reality, rather than from the universe/world/reality to maths. That is, the maths came first and only after was it applied to the universe/world/reality.

The question we must now ask is this:

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

This isn’t to only to state that maths can describe “things” – it’s to say that “things [literally] are numbers”. But what does that actually mean? As with Max Tegmark (who endorses this position), we can ask it the statement “All things are numbers” is to be taken poetically or literally. Taken literally, it hardly makes sense. Taken poetically, it still requires interpretation.

One interpretation of both Tegmark's and the Pythagorean positions is that if things literally are numbers, then it’s no surprise that string theory is on top of things when it comes to describing reality. I mean that in this sense:

i) If things are numbers,
ii) and numbers describe things (which are numbers),
iii) then numbers are describing numbers.

So we never escape from the world of numbers; which is, I suppose, precisely the result which Pythagoreans want. Yet we still don’t really know (for sure) what the statement “All things are numbers” means.

To change tack a little.

The physicist John Archibald Wheeler provided the best riposte to Pythagoreanism in physics. We're told 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.

Then Steven Hawking 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 science writer Kitty Ferguson offers a (possible) Pythagorean answer to Hawking's question when she says 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, “breathes fire into the equation [to] make a world”?

To return to string theory.

Kaku himself puts what can be seen as the extreme place which string theory finds itself in. He writes:

My own view is that verification of string theory might come entirely from pure mathematics, rather than from experiment.”

This is hard to even understand. That may because I’ve read too much philosophy and find the use of the words “verification” and “mathematics” together odd. That isn’t a problem if Kaku means “proof” by “verification”. Whatever the case may be, it’s clear that he doesn’t mean observational or empirical means of verification. After all, Kaku himself finishes off with the words “rather than from experiment”.

So is string theory is about the physical world? Yes?

It's true that all physics employs mathematics. However, such physics is still about the physical world. Therefore how can the “verification of string theory [come] entirely from pure mathematics”? (Note the termpure mathematics”, not “applied mathematics”.) How would that work?

Despite all of the above, string theorists haven’t got a complete hegemony. Kaku happily acknowledges this and cites various cases. For example, the chance meeting of the theoretical physicist John Schwarz and Richard Feynman. Kaku writes:

String theorists became the butt of jokes. (John Schwarz remembers riding in the elevator with Richard Feynman, who jokingly said to him, ‘Well, John, and how many dimensions do you live in today?’”

Feynman’s witty remark didn’t necessarily mean that he was against string theory. After all, this exchange occurred in the early days of string theory (Feynman died in 1988). In addition, Feynman's “sum over paths” is hardly a walk in the park.

Of course we need to establish whether or not there’s a strong connection between the fact that string theory is highly mathematical and that corresponding fact that string theorists believe some pretty outlandish things about the universe (or multiverse).

The Beauties and Symmetries of String Theory

Much is made of the fact that many mathematicians and physicists stress the “beauty” and “elegance” of their theories. In terms of string theory, perhaps these aesthetic values may provide string theory's main appeal.

Kaku explicitly defines (mathematical) beauty thus:.

When physicists speak of 'beauty' in their theories, they really mean that their theory possesses at least two essential features: 1. A unifying symmetry 2. The ability to explain vast amounts of experimental data with the most economical mathematical expressions.”

Then Kaku quotes the physicist and astrophysicist Joel Primack saying as much. (Though he''s talking about inflation, not string theory!) Primack said:

No theory as beautiful as this has ever been wrong before.”

It would be an empirical fact if that were true. However, Lee Smolin, for one, has warned physicists about paying to much homage to mathematical beauty. (Roger Penrose has also warned about this in his Road to Reality, chapter 34)

Smolin quotes string theorists talking about the beauty of the theory in the following:

“... 'How can you not see the beauty of the theory? How could a theory do all this and not be true?' say the string theorists.”

Smolin is at his most explicit when he also tells us that “string theorists are passionate about is that the theory is beautiful or 'elegant'”. However, he says that

“[t]his is something of an aesthetic judgment that people may disagree about, so I'm not sure how it should be evaluated”.

Smolin concludes:

“In any case, [aesthetics] has no role in an objective assessment of the accomplishments of the theory.... lots of beautiful theories have turned out to have nothing to do with nature.”

Perhaps Smolin is going too far here. Surely it's the case that aesthetics has at least some role to play when it comes to theory-choice. And who's to say that whether a theory is elegant or not – at least in some sense - isn't itself an “objective” issue? Whether something is simpler than another theory is surely an objective fact. It's whether or not such simplicity can also be directly tied to elegance, beauty and truth that's the issue here.

In addition to that, as evolutionary psychologists and cognitive scientists have told us, human beings have an innate need for both simplicity and explanation – sometimes (or even oftentimes) at the expense of truth. Thus, in the case of string theory, we may have a juxtaposition of the psychological need for simplicity and explanation along with highly-complicated and arcane mathematics.

So perhaps that highly-complicated maths is but a means to secure us simplicity and explanation. That is, the work done towards simplicity and explanation is very complex and difficult; though the result – a theory which is both simple and highly explanatory – evidently isn't.

All this means that the statement “beauty is truth” may not itself be a truth. Or at least it may not always be applicable to every mathematical or physical theory.

And just as the two-way direction of world-to-maths or maths-to-world has been discussed, so we have a similar idea with beauty. This is how Kaku puts it:

When you come up with a theory, you fall in love with the beauty the simplicity and elegance of it. But then you have to get a sheet of paper and pencil and crack out all the details. Hundreds and hundreds of pages. Because you have to prove it.”

Here Kaku (or other physicists) firstly notes “the beauty, simplicity and elegance” of a theory. And only then does he “get a sheet of paper and pencil and crack out all the details”. So beauty is first spotted (as it were), and only then does the physicist needs to “prove it” - that is, show that beauty is also truth/correctness. But, again, the beauty of a mathematical theory comes first. Thus beauty in and of itself (as it were) entails (to use a strong word) truth.

Yet talk of beauty by physicists is very off-putting because their views of beauty don't really coincide with the layperson's – except when it comes to symmetry and simplicity. But even here the way that physicists use the words “symmetry” and “simplicity” will not strike many chords with many laypersons. But perhaps that simply doesn't matter.

Kaku offers us a very specific case of beauty-vs.-ugliness in the following account of Maxwell's equations:

Maxwell's equations... originally consisted of eight equations. These equations are not "beautiful." They do not possess much symmetry. In their original form, they are ugly. ...However, when rewritten using time as the fourth dimension, this rather awkward set of eight equations collapses into a single tensor equation. This is what a physicist calls 'beauty,' because both criteria are now satisfied.”

Clearly, beauty is being strongly connected to symmetry. And, of course, string theory and M-theory offer us much symmetry. In architecture and music, symmetry can be very important. Indeed they may partly constitute beauty. What about simplicity? That depends. Aesthetics is a tricky road to go down if one isn't an artist or aesthetician.

String Solutions and the Multiverse

Let's give one specific example of how the physics of string theory grew out of the mathematics of string theory.

In the mid-1990s, billions of solutions of what are called the string equations were discovered. The important and relevant point to make about this situation was that all those billions of solutions “corresponded to a mathematically self-consistent universe”. This had a concrete effect, then, on physics itself and especially on string theory. As Kaku tells the story:

The bewildering numbers of string solutions was actually welcomed by physicists who believe in the multiverse idea, since each solution represents a totally self-consistent parallel universe.”

Interestingly enough, Kaku continues by saying that

it was distressing that physicists had trouble finding precisely our own universe among the jungle of universes”

Put that way, it’s as if this “jungle of universes” grew solely out of the maths. That is, firstly we had billions of string solutions, and only then we had “billions of self-consistent parallel universe[s]”. This raises the question as to how many physicists believed in the multiverse theory before the mid-1990s (i.e., when these string solutions were discovered). The way that Kaku puts it (which may be a simple grammatical misreading) is that there were physicists who already believed in the multiverse - and then these string solutions came along to back up their prior beliefs. That is, Kaku wrote that “[t]he bewildering numbers of string solutions was actually welcomed by physicists who believe in the multiverse idea”. Does this mean that before the 1990s physicists had neither mathematical nor physical reasons to believe in the multiverse? Or did previous mathematical tricks (which helped bring the multiverse into existence) already exist before that time?

So what do physicists mean by “beauty”? Here Kaku helps us out again. He writes:

To a physicist, beauty means symmetry and simplicity. If a theory is beautiful, this means it has a powerful symmetry that can explain a large body of data in the most compact, economical manner. More precisely, an equation is considered to be beautiful if it remains the same when we interchange its components among themselves.”

Of course string theory and M theory contain many symmetries. So does that mean that string theory (or string theories) must be beautiful? However, despite these positive words, Kaku does go on to say the following:

Symmetries then encode the hidden beauty of nature. But in reality, today these symmetries are horribly broken. The four great forces of the universe do not resemble each other at all. In fact, the universe is full of irregularities and defects...”

Despite the fact that Kaku makes much of symmetry and supersymmetry, he’s still happy to acknowledge “at present there is absolutely no experimental evidence to support it”. Then again, he does conclude by saying that

[t]his may be because the superpartners of the familiar electrons and protons are simply too massive to be produced in today’s particle accelerators”. (This was written in 2005.)

Is this is a clear case of supersymmetry wagging the dog (as it were)? Nonetheless, one can see the “beauty” of supersymmetry and the consequential urge to embrace it. Kaku sells it like this:

When one adds supersymmetry [to the Standard Model], however, all three forces fit perfectly and are of equal strength, precisely what a unified field theory would suggest.”

But then comes a warning sign:

Although this is not direct proof of supersymmetry, it shows at least that supersymmetry is consistent with known physics.”

So here we have mathematical and physical consistency again. Like the possible worlds of philosophers, all these things are indeed logically possible. Zombies and trees with a sense of humour are also logically possible: but are they actual?

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