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Source: How should logarithms be taught? All bolds are mine.

Those who criticized this view tended to think that I was advocating pure rote learning rather than understanding. Actually, I was suggesting that a true understanding of a sophisticated concept such as the exponential function involves letting go of the intuitive meaning (once it has served its purpose of telling you the rules you want the function to satisfy) and using the defining properties instead.

Behind that suggestion is a more general claim, which is that mathematicians greatly underestimate the extent to which they think syntactically rather than semantically. [...]

Do the Linguistic definitions below relate to these terms' meanings in math?

Source: An Introduction to Language (10 ed., 2013. But $\exists$ 11 ed.)

[p 578:] semantics 
The study of the linguistic meanings of morphemes, words, phrases, and sentences.

[p 582:] syntax 
The rules of sentence formation; the component of the mental grammar that represents speakers’ knowledge of the structure of phrases and sentences.

  • I think he's saying mathematicians think more in symbols than they think; while they assume they mostly think in terms of concepts, Gowers says they underestimate how much of their thought process is formal – Maxime Ramzi Dec 18 '17 at 07:01
  • Syntax = symbols manipulation. Semantics = formulas interpretation. – Mauro ALLEGRANZA Dec 18 '17 at 07:12
  • I think that anyone who mastered reindexing of multiple sums, Fubini's theorem and differentiation under the integral sign, or just a meaningful amount of dirty&effective tricks is aware that he/she is thinking syntactically rather than semantically :) – Jack D'Aurizio Dec 18 '17 at 07:13
  • @JackD'Aurizio Sorry for any misunderstand, but were you hinting towards me by 'anyone'? –  Dec 18 '17 at 08:00
  • @Canada-Area51Proposal: no, it really was a generic anyone. – Jack D'Aurizio Dec 18 '17 at 08:33

1 Answers1

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The linguistic meaning of syntax and semantics roughly corresponds to the mathematical understanding.

Syntax is the symbol-shuffling operations in mathematics. This includes substitution. In the context of logarithms, there's the rule that $a^y = x \implies y = \log_a(x)$. If we use this rule by substituting different numbers into roles of $a, x, y$, we're thinking of logarithms syntactically.

Semantics is the underlying meaning of the equations. This includes object representations. So, when we're thinking about logarithms, we think about the exponentiation number. Like, how $\log_{10}(100) = 2$, which means we exponentiated $10$ two times to get to $100$. The point being, rather than defining a logarithm as an inverse, or using rules, we consider it meaning-first.

These words have much more specific definitions in mathematical logic, where the objects studied are truth values rather than numbers. Syntax is like proof rules, and semantics is the actual truth value, interpreted under a model.

Larry B.
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