I have to say, Edward Frenkel's Master Class "
Mathematics: The Language of Nature" was something of a disappointment. Things were hinted at, but never explained. In many ways, it also seemed more of a politics (his pet project) and economics (his book) inspired talk than one trying to convey an understanding of really important mathematical concepts. Honestly, the only thing I came away with was the basic notion that there were certain fields of mathematical inquiry that had names, and that there were hints that there were connections between them. He tossed up a bunch of concepts (e.g. group theory) and failed to provide any insight into how they worked and why (more of a "name dropping" sort of thing than an educational lecture). Overall, I'd give this one failing marks. The questions they asked were also somewhat disappointing and not very helpful in forming further understanding of the issues.
Is mathematics something inherent to nature, or is it a purely human construct? Why do you think math is so effective in describing the physical world around us? Do you think math exists by itself without physics?In some ways, I think there is a selection bias present in the notion that abstract mathematical concepts have often led to physical representations: we see many success stories and take those successes to be indicative of an underlying connection. On the flip side, mathematics explores the infinite ways that connections and patterns and representations can exist. It stands to reason that everything physical are made up of those same elements, so there is at least that fundamental similarity. One thing that always struck me is how solving a mathematical problem can provide multiple answers, only one of which can possibly correspond to the physical reality it is linked with. A neat example I saw just the other day is Susskind (on YouTube) was solving a simple quadratic equation to arrive at the form of an unknown function. Without questioning, he took the positive root and ignored the negative root. When questioned by a student, he flailed about for a few minutes and finally realized that if you plugged the positive root into the equation, you got, well, an equation (literally, x = x) that made physical (physics) sense; but if you plugged in the negative root, you got a broken/un-useful (but true) result (that x = -x, which is only true if x = 0... not helpful if we happen to be living where x = 1). So, he was able to select a particular answer of many (well, two in this simple case) because it corresponded with the physical reality we experience, and rejected another mathematically valid answer because it did not. I think math does exist by itself without physics, but constraining math with physics allows us to select those tools or techniques that resonate with how our universe is manifest. Requiring that discoveries in math somehow eventually be tied in some ways to our physical reality empoverishes the field in the same way that suggesting that all art must be commercially viable empoverishes creativity in that field (or that all science must have a business application, etc.).
We've seen that one object can be "more" symmetrical than another, based on the number of rotational symmetries it has. A circle has infinitely many of these symmetrical transformations, but we also learned that a sphere does as well. Can we say that one of these objects is more symmetrical than the other?From what Frenkel said, a sphere can rotate about its axis like a circle, but then we can also choose where we put the rotational axis in a sphere. By contrast, in a circle the axis can only be perpendicular to the plane of the circle and in its centre. Thus, a sphere is more symmetrical than a circle.
It took mathematicians 300 years to solve the mystery of why a general algebraic solution did not exist for polynomials of degree 5 and higher. The discovery of a bigger framework was required to push the boundary of our understanding. Do you think a unification of mathematics or physics requires yet another reformation of our current framework?I would tend to believe that mathematics is not a dead subject (meaning that it is complete as it stands), and there will therefore be grand discovery after grand discovery for as long as people wish to pursue the subject. I suspect mathematics will be like physics in that as known realms are unified, the work done to achieve that unification will uncover little inconsistencies or hints of other realms that will grow into full fields of studies of their own that will require unification, and so on. Given the abstract nature of math, it would seem to me that it is an infinitely fractal subject that can be explored forever (although the main branches will be thoroughly explored and understood over time even as new branches are added). So the trite answer is yes, another reformation is needed. However, I would suggest that once that is achieved, there will be demand for yet another reformation after that, and then after that, as we use the new understandings we generate at each step to clarify what lies outside of explanation at each moment.
Is unification something we can potentially reach? Do you think it is possible to find a "theory of everything," or will we inevitably continue to look for a deeper fundamental understanding?I think that the question is non-sensical in that the word "everything" changes its meaning over time. If you never leave your house, "everything" has a different meaning to you than to someone who has spent a year in orbit around the Earth (or has been to the moon). Could we find a theory of everything we know so far? Probably... or at least we can learn to understand why some things truly cannot be unified with other things (there is Gödel's fundamental principle of incompleteness as well to consider when we ask these questions). But as we learn to work with what we know now, new discoveries will be made such that "everything" becomes a larger entity, and we will struggle then to find new meanings and connections. Poking at physics for a second... it may be that despite mathematical suggestions to the contrary, there really is no such thing as a magnetic monopole (at least not in any way that can have a phenomenological impact). Similarly, perhaps gravity is something fundamentally different from the other three force carriers and there will never be a unification at our current level of understanding (not to say that there won't be some deep underlying principle found that is responsible for all four forces, and matter itself, but unification may elude us until that common principle is discovered, should it exist).
