Definitely a skill almost everyone possesses today. Everyone is used to think in terms of operations and effects of operations, units of information, be it bits or qubits etc. I find this subject fascinating and worth exploring. There is another part of this subject, less explored… I feel that its origins are to be traced back to Newton or Leibniz (whoever was first) and their way of looking at the universe. What they observed was that small changes can be made arbitrarily small and one can gain information about how things change by looking at very small variations and eventually integrate them. The stress here is on “very small”… One can think in nature about things that are arbitrarily small but one cannot represent something arbitrarily small on a computer. Then people started thinking about how “small” changes could affect some of of the properties described before. This is how people gained an intuitive image about what we call today “perturbation theory”. This stuck to human minds. It stuck so deep that it is very hard to use any of the formulations of today (be it physics, computer science, mathematics) to describe anything that evades this paradigm. Any tentative to evade this way of thinking remained more or less “in the dark”… the two options are : either using lattice models (assumed by some over-enthusiastic people as a “property of nature” at a rather strange moment in history, see “the universe is a lattice” idea)  or entering the idea that a perturbative description valid in some area can be made dual with another perturbative description valid in another area (see strong-weak dualities). Both ideas are based on the same way of thinking : something must be “small”, either the lattice spacing or the expansion parameter(s) of the theory one choses to calculate with… One may ask if this method is sustainable and the answer is obviously NO: while a duality can make some regions of a theory that are not reachable from one theory, reachable in the dual theory it misses all the other regions one may arrive at (or not) using other unknown “dualities”…

Nevertheless, if there is some science that allows one to make statements valid in a general way then that is mathematics. It gave birth to a domain that is focused on precisely the opposite of what people were thinking in the natural sciences: what are the properties that do not depend on “small variations”? Or, another way to ask the question: what happens if we don’t actually care about the path you take in order to get from one place to another and focus only on the fact that the two places are reachable if one starts from one of them? These are probably the first more interesting questions a mathematician asked… They were brought into physics by several authors during the 1980, mainly Atiyah, Singer, Grothendieck, Bott, Witten, and many many others, willingly or less so…

Still, the way of thinking of physicists even today is mostly bound to perturbative analysis. Whatever goes beyond perturbation theory is still bound to its imprint it had on the human mind. The change of paradigm that will free humans of their perturbative way of thinking did not appear yet and String Theory is the ultimate victim of this way of thinking.


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