What is the formal definition of a mathematical statement? We can all agree that the statement "Humans are apes" is not a mathematical statement, and the statement "4 is a prime number" is a (false) mathematical statement. But there seems to be no formal definition that sorts mathematical statements from other statements. Has anyone come up with such a definition?
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1Mathematical claims are those about abstract objects with formal properties. Generally, a claim is mathematical if it can be expressed as a proposition in some formal axiomatic system, such as ZFC.– causativeApr 17, 2021 at 6:18
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4No "formal" definition at all. Mathematical statements are those contained into mathematical books.– Mauro ALLEGRANZAApr 17, 2021 at 10:05
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Mathematical is something that relates to mathematics. Can mathematics say something about that statement? It is mathematical. It can't? Then it isn't.– user31740Apr 17, 2021 at 10:11
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2A statement is mathematical if it belongs to a mathematical theory, "4 is a prime number" is such because it belongs to arithmetic. But formalize the biological concepts involved and you can make "humans are apes" into a theorem too. What makes a theory mathematical is controversial, see definitions of mathematics, but something with algorithmic rules for forming expressions and making inferences about them will be almost universally accepted as such today. In practice, just sketching such rules will suffice.– ConifoldApr 17, 2021 at 10:27
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In the correct context, it is mathematical if surrounded by dollar signs. However, this stack does not turn on the equation writer.– BillOnneOct 10, 2022 at 19:26
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3 Answers
Mathematics is a language and a tool.
As a language, it is specifically a formal language, which essentially means that it allows communicating ideas in a specific language, strictly determined by a set of rules.
(Answer to your question:) a mathematical statement is just an statement which conforms to the rules of the mathematical formal language.
As a tool, mathematics is a formal system, that is, a system of axioms and concepts, which allows logical calculus in order to produce further conclusions from such axioms and concepts.
I suggest you to take a look to the Goedel's incompleteness theorem, which determines an amazing mechanism to provide a complete description of all possible mathematical axioms, and the problems that would derive from such task. That gives a good grasp of the structure of a mathematical statement, which is determinant to Goedel's conclusions.
"Humans are apes" can be distilled into a mathematical statement if you have a Turing test like criterion to arrive at certain abstract essentials to define or identify humans and apes respectively, then to state "Humans are apes" you just need to list all the matching equalities between them one by one, even possibly infinite amount of such equalities.
As many people already commented or answered above, it's not easy to strictly define what's a math statement since math covers so many fields and categories. Similar to Ted Wrigley above, I just would like to replace his measurement with quantification. Mathematics is a particular formal language constructed around abstract rules about quantification over relations which can be reduced to logic (such as second order logic which admits such quantification). In general we can fairly quantify anything; measuring things is often more problematic. For example in an engineering team, you can easily quantify all kinds of metrics to represent its members productivity, but what to measure as a definitive yardstick to judge each member or the whole team can be very controversial...
answered Apr 18, 2021 at 4:30
A mathematical statement is any statement that measures relationships. 1+1 = 2 and 12+12 = (√2)2 say the same thing, but measure different features of the world.
Any natural language measurement statement can be translated into mathematics by removing specificity, and any mathematical statement can be instantiated in natural language by supplying specificity. Statements that do not involve measurement — declarations, identifications, analogy and metaphor, dictates, social conventions, etc. — have no mathematical representation. In Wittgenstein's (later) worldview, mathematics is a particular language game constructed around abstract rules of measurement, that has no particular bearing on other language games.
answered Apr 17, 2021 at 16:06
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Just some note here, your key concept "measurement" here seems need to be emphasized to be interpreted more generally in addition to usual geometric or quantitative measurement, such as topology can be interpreted as measurement of "betweenness", group can be interpreted as measurement of "symmetry"... Apr 18, 2021 at 2:03
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Measuring???? For any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. Apr 18, 2021 at 10:35
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@MauroALLEGRANZA: Do you agree or disagree with the statement: "every practical application of the Brouwer fixed-point theorem (linked above) is an act of measurement". The entire point of that theorem is to guide us discovering where f(x_0) = x_0. Much of mathematics is theoretical investigations about the rules, limits, and properties of systematic measurement, but in a real sense modern topology and calculus are just elaborations of ancient Greeks scratching out geometric relationships in the dirt. Apr 18, 2021 at 15:23
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@DoubleKnot: Yeah, honestly, my entire answer needs a hell of a lot more interpretation and explication: likely at book length. (sigh...) I want to wait and see what comments pop up before I consider revisions and expansions; that will help guide what needs to be expanded. Apr 18, 2021 at 15:27