Mathematics are doing a very odd usage of syntax and semantics. Let's take a wikipedia page as an example : https://en.wikipedia.org/wiki/Intuitionistic_logic

Here we have a syntax which is given, and only after the semantics and actually even multiple semantics ! The cognitive process seems very strange to me, like inverted.

When I want to say something, when I want to say that I like cherries for example, well I start with a semantics, the truth I want to say, then I chose a language, let's say english, or my mothertongue, and finally I try to find words in this language that will encode what I want to say, may be "I like cherries" or "I like cherries very much". Many ouput syntaxes are possible but there is only one semantics as the input of my cognitive process.

It seems to me quite strange to start with a sentence like "I like apples" and then to ask but what could it means ? Oh if "apples" means chocolate it means that I like chocolate, I found a new interpretation for my syntax ! But actually this is how modern mathematics are offered.

It is like the syntax was the most important part in mathematics, whereas I see the syntax more as a concrete mean to communicate some theoretical semantics, which would be the most important part.

Can someone explains this "inversion" ?

  • 1
    If you want to communicate semantics first you need to coach it into some other syntax that the target audience is already familiar with, typically, the vague natural language. Since mathematics is about reducing vagueness syntax is the most important part of it, really, the only part. "Semantics" is just another syntax, as you can see from model theory where the syntax of ZFC is used as semantics for other theories. How the "ultimate" vague semantics, i.e. the natural language, is mastered and used is not a suitable subject for mathematics given its purpose.
    – Conifold
    Commented Jul 19, 2022 at 18:19
  • If semantics is not so important, what is the point of reducing the vagueness of its syntax ? I feel this like art which is no longer speaking about anything else than art.
    – François
    Commented Jul 19, 2022 at 18:34
  • It is not unimportant - it is misconstrued, as something separate from syntax. The reason you feel this way is likely because you are overwhelmed by an unfamiliar manner of speaking, not because it does not speak to everything else. That will pass with reflecting on why this manner is unavoidable for speaking better (with more precision), and experience. It feels the same when learning any new language.
    – Conifold
    Commented Jul 19, 2022 at 19:15
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    Your linked WP source mainly targets people who want to learn and become familiar with how to read a new type of logic, so such inverted sequence is optimal. If you want to write some sentences of such language or create your own type of logic, then it's not inverted per your language... Commented Jul 19, 2022 at 19:46
  • 1
    Yes I can understand, even if I think that the better way to read is to write.
    – François
    Commented Jul 19, 2022 at 22:36

1 Answer 1


You're missing some points here, namely the notion of intended interpretation and axiomatization. Often one works from model to axioms. However the surprise can be that the axioms may have also have a different model/interpretation than intended. So, if you e.g. read about how (axiomatic) set theory was devised, it was more like an iterative process, with "trial and error" as far as axioms went etc. More on that here on the development of axioms for arithmetic, as by then the implication of non-standard models was already understood.

As you used intuitionistic logic as your motivating example, the intended interpretation for intuitionistic logic (when devised) was the BHK interpretation.

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