> I guess a very large majority of people would still think that math is the rational, systematic account of what is ("real world"), but Atiyah seems to say that from an inner-mathematical perspective, the purely formal conception of mathematics prevailed. Algebra was the "Faustian offer" handed over to mathematicians: in exchange for the formidable machine of symbolic reasoning, we would have to sacrifice the meaning of what we are dealing with, at least temporarily.

I am not sure that this has been a "good thing" for modern mathematics. While symbolic logic is definitely a necessity, it has been carried too far in as much as most folks are unable/find-it-difficult to model "real world" phenomena. Abstraction proceeds from the concrete to the general but if one loses sight of this link all symbolic manipulation is mere playing games without any understanding.

V.I.Arnold in his essay On teaching Mathematics makes this very point - https://www.math.fsu.edu/~wxm/Arnold.htm

Excerpts:

The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter-examples which would prevent unjustified extension of our observations onto a too wide range of events.

As a result we formulate the empirical discovery that we made as clearly as possible. After this there comes the difficult period of checking as to how reliable are the conclusions .

At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be "absolutely" correct and are accepted as "axioms". The sense of this "absoluteness" lies precisely in the fact that we allow ourselves to use these "facts" according to the rules of formal logic, in the process declaring as "theorems" all that we can derive from them.

It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result.

In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions ("proofs"), the less reliable is the final result.

The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called "the inconceivable effectiveness of mathematics in natural sciences" (or "the Wigner principle").

"The subtle poison of mathematical education" (in F. Klein's words) for a physicist consists precisely in that the absolutised model separates from the reality and is no longer compared with it.

nor discussing the danger of fetishising theorems are to be met in modern mathematical textbooks, even in the better ones. I even got the impression that scholastic mathematicians (who have little knowledge of physics) believe in the principal difference of the axiomatic mathematics from modelling which is common in natural science and which always requires the subsequent control of deductions by an experiment.

Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators.