Hope a big prize is soon awarded for the use of string theory’s AdS/CFT to solve problems in condensed matter physics or QCD string interactions! Maybe when it starts getting prizes for applications, it will be seen in context more clearly, and will need less hype for unification and quantum gravity applications in popular media.

Physics is fundamentally flawed for observing the vacuum is isotropic to photons and assuming the massed sector in kind. Quantized gravitation theories require supplementing Einstein-Hilbert action with a parity-violating Chern-Simons term. Do left and right shoes vacuum free fall identically? Load an Eötvös balance with chemically and macroscopically identical solid single crystal quartz test masses, enantiomorphic space group P3(1)21 opposed by P3(2)21, right- and left-handed screw axes respectively. Is there a net non-zero signal? There’s your problem.

I’m not exactly a mathematician, but I’ll give it a go! Keep in mind that I typically look at math from a formalist point of view; a Platonist would probably answer differently than me.

(I apologize profusely for the length of this post—I have to do it now, because you might not make it to the end!)

To me, asking, “What are the fundamental building blocks of mathematics?” is like asking, “What are the fundamental building blocks of art?” People create art using whatever they have lying around—no material or technique is more fundamental than all the others. And people’s idea of what art is is changing constantly. A European Renaissance artist probably wouldn’t see scrap metal, dirt, or cheap consumer goods as building blocks of art, but a modern European artist might feel differently.

You said, “I was thinking it all begins with the notion of a set and that the notion of a function also plays a part.” In case you don’t know, the fact that this idea is so common right now has a lot to do with our historical context. At the beginning of the 20th century, a lot of mathematicians—most famously, Bertrand Russell and Alfred Whitehead—got interested in trying to derive all of mathematics from a few axioms and rules of inference. They eventually decided to use the notion of a set as the seed from which all of mathematics would grow, and the framework they came up with is now known as axiomatic set theory. It’s true that most of modern mathematics can, in principle, be built up from axiomatic set theory, but I wouldn’t say that sets are therefore the fundamental building blocks of math. Trying to build something like the formula for the volume of a sphere out of axiomatic set theory would be like trying to build a sculpture one atom at a time. To me, saying that sets are the fundamental building blocks of math is like saying that atoms are the fundamental building blocks of art. It may be true in some sense, but it totally misses the point!

Moreover, axiomatic set theory is by no means the only possible foundation for mathematics. For example, there’s a brand-new subject, called category theory, that’s becoming more and more important in many areas of math. Recently, people have started talking about the Elementary Theory of the Category of Sets: an effort to build axiomatic set theory out of category theory. At first, this looks like a step down to a deeper level of description; if sets are the atoms of math, maybe categories are the subatomic particles. But here’s the interesting thing: if you take an introductory class on category theory, the teacher will probably start by describing categories in terms of sets! In fact, my gut feeling is that category theory can be built out of axiomatic set theory (because category theory is kind of like abstract algebra, and abstract algebra is a pretty important area of modern math, and most of modern math can be built up from set theory). So here we have two alternative “foundations”… and each foundation seems to be built on top of the other!

To conclude, I’ll give you my answer to a question you didn’t ask. By inquiring about the “building blocks” of mathematics, you seemed to be asking for tangible objects—things like sets and functions, paints and brushes, axioms and rules of inference, scrap metal and welding torches. If you want to get to the heart of mathematics, you should be looking for intangible things: truth and beauty, pattern and asymmetry, imitation and creativity, proof and intuition. It’s things like these that are the seeds from which all of mathematics grows.

I don’t have anything more profound (or even mundane) to say beyond my earlier remark. It is not an issue I can say usefully or interestingly anything about at this time.

-cvj

Interesting!

I’m sorry to disappoint, but I don’t even know where to begin to answer this question. Does it begin with geometry? Algebra? Topology? Is it a meaningful question? Don’t know. Ask a mathematician perhaps. I don’t even know if it is easier or harder a question to answer than for physics.

Best,

-cvj

What are the fundamental building blocks of mathematics in your view?

Is this not a more fundamental question than ‘what are the fundamental building blocks of nature?’ given that the tools you use to describe nature are mathematical and you ‘need not confuse the tools with the things we are trying to describe with them.’ as you say above?

Thanks,

I don’t think you can have read my post carefully. In several places I stated that there are central principles to be found in both “applied” physics and that which you consider to be the more noble quest of uncovering “reality”. Whatever drives you to whatever physics you choose, focusing on one and ignoring the other can be a major mistake, from the simple pragmatic stance of finding good sources of ideas, techniques, and mechanisms. It is all clearly stated in the post. I don’t think I need to repeat it all. Please have a look.

Best,

-cvj

You may, at least in principle, discover what this place really is, how it came to be and where it is going. Those are one of the most important questions there are. (I am not claiming you can under them completely but every insight counts here)

For example the discovery of spacetime is one of the most profound discoveries of all time, it has changed our understanding of reality forever.

Working on applied physics is a completely different thing, there are some interesting problems but hardly anything as profound.

Another point is that it’s a good thing that the reductionist frontier attracts only a tiny fraction of scientists to work on it, because the necessary experimental efforts to explore it are impossible without the vast supporting technology that had to be developed, and the rich technology-enabled societies that can afford the resources.

A very nice essay, thanks. I summarized some thoughts on the limits on reductionism here. You might also find this paper interesting. Best,

B.

-cvj

Careful, you’ll start looking like Diana Ross.

Interesting thought about the learning curve. I’m not so sure -once the dust settles- that it is any steeper, relatively speaking, than it ever was. I think that once the toolbox is streamlined (as a result of the kind of research going on now) and organized (put the sharp cutting tools all together over there, the blunt breaking tools over there, the measuring tools together over there) and an instruction manual written, it’ll seem no more challenging than learning new things in any new field.

Right now it is daunting because nobody knows all the tricks, what are the red herrings, and so forth.

Best,

-cvj

That’s not the issue. To focus on some narrow aspect of one field or another to make one seem more interesting than another is a trivially straightforward game to play and utterly misses the point of what I’m talking about.

Cheers,

-cvj

I see what you are saying, *but* to be frank I, and many others, are interested in [for example] the RHIC *only* because of the dazzling possibility that what they are doing there has something to do with black holes in five-dimensional asymptotically AdS spacetimes. If there were no such connection then I’m afraid RHIC would be just another tedious gadget to me. So the dreaded physics pecking order is still lurking in the background. But who cares really?

“(Remember, you heard it here first.)”

I will indeed. Non-professional readers should be alerted that Prof Johnson here is actually the True Father of this whole field*. I got a bit of a jolt recently when looking at his book on D-branes — it foretells all this marvellous “applied string theory” story, and it was written ages ago. So our host here has quite a record as a prognosticator in this line……watch this asymptotically AdS space!

* I hereby prognosticate that he will deny this.