Does the Peano axiom of reflexivity ∀x(x=x), in that the statement 'x=x'is a tautology and contains no information, imply that mathematics lacks ontological reality? By this I mean that if the 'building blocks' of mathematics are empty of information from their own frames of reference, does mathematics then provide truth only within an ontologically relational framework?
asked Feb 1, 2017 at 14:54
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Still unclear...– Mauro ALLEGRANZAFeb 1, 2017 at 14:55
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1The (reflexivity) axiox for equality is not "tautological" in the sense you are alluding (i.e. trivial). It is "trivial" but needed in order to specify the necessary properties of "equality" and we cannot formalize a mathematical theory without equality.– Mauro ALLEGRANZAFeb 1, 2017 at 14:56
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Having said that, arithmetic is not deprived of "ontological content" (assuming that we know what it means) : Peano axioms for arithmetic codify the "basic" property of natural numbers.– Mauro ALLEGRANZAFeb 1, 2017 at 14:58
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1you might want to look into HoTT, where an equality is a type, and refl is its constructor. which means refl is not an axiom– user20153Feb 1, 2017 at 18:55
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1On the complex philosophical issues regarding the ontology of numbers, see Phil of Math with many linked entries.– Mauro ALLEGRANZAFeb 1, 2017 at 20:13
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1 Answer
The axiom of reflexivity, specifying that equality is reflexive, simply helps define one aspect of the '=' symbol. This helps make a connection between well-formed formal strings that use that symbol and our personal conception of what equality should act like. It only seems boring and tautological because the concept is obvious, but the axiom is needed to operate on the symbols usefully. The axiom contains quite a bit of information ('usefulness' might be more appropriate) because without it, all sorts of theorems about natural numbers could not be proved.
I don't know what you could mean by 'empty of information in their own frame of reference'. You're using words that don't have any technical relevance to axioms and proof theory. If you're using them non-technically, you'd really have to explain what you mean by 'information' and 'frame of reference' as they refer to the mathematical system.
But does 'mathematics then provide truth only within an ontologically relational framework'? Whatever you think about those words together, a yes or no answer to it will only justify some hidden personal definitions to make the answer right.
But to jump from the axiom that equality is reflexive and that it feels tautological all the way to a statement about all of mathematics and reality (and ontology) is just perverse. The axiom is included to make proofs work (if you do some examples you'll see where the axiom comes in useful). If you want to draw the earth-shattering consequence that there is no absolute truth, the seeming tautology of reflexivity in arithmetic is not the place. (on the other hand, truth is undefinable inside arithmetic, but that's not the truth you're thinking of)
To help with this, is there anything special about reflexivity axiom? Aren't symmetricity and transitivity also tautological? What about the other axioms? Axioms are often very boring because they should be intuitively non-questionable (in order to trust that the proof system is proving things that fit with your intuition).
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There are axioms that define numbers (reflexivity and transitivity too, though are they inferred?) and axioms that define how those numbers can be operated on (arithmetic ones). If the axioms that define numbers provide 0 information about what those numbers are, where is their 'reality' drawn from? Mathematics describes the world accurately, so it is useful. But if experience showed that 1=1 for all objects other than oranges, (where 1 orange = 2 oranges), the axiom would not be intuitive. I'm not saying throw them out btw, just wondering how they are 'real', beyond us saying 'they are real'. Feb 1, 2017 at 19:50
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1You are getting all worked up about words that are being used both technically and loosely at the same time (real, information, axiom infer). Also you have a lot of issues here to be addressed and a discussion may be more useful to you (and a Q&A environment does not fit well with that). It seems like you need a firmer grounding in both practical logic (manipulating axioms and proofs) and in the varieties of philosophies of mathematics (e.g. platonism fictionalism, social realism)– MitchFeb 1, 2017 at 20:41
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But I will try in comments here for some issues. 1) reality - can you hold 2 in your hand? You can hold 2 apples or 2 oranges, but what about 2 itself? And is holding necessary for realness? 2) What is the difference between 'information about what a number is' and 'what their reality is'? 3) Suppose x=x works for everything in the 'real' world except for oranges. Then maybe you don't want to bother with oranges in thinking about '=', or maybe you'll want a slightly different axiom (that takes into account oranges).– MitchFeb 1, 2017 at 20:46
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Math is what philosophy wishes it could be, where all terms are defined and other people can repeat your thought steps exactly to get the same conclusions. Maybe the Peano axioms aren't the best, or maybe you're worried that those axioms work with 'real' 'objects' other than the natural numbers. Mathematicians (logicians specifically) have investigated if there are other objects than the naturals that satisfy the Peano axioms but are distinct (there are. They are called non-standard integers).– MitchFeb 1, 2017 at 20:52
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...They act like naturals (follow the same axioms), but have some properties, not touched on by the axioms, that are different, in fact non-intuitive as you mention. But what is reality then? Wait... mathematical thinking is inspired by thoughtful abstractions of experience (5 fingers...five toes... holy crap '5'!) ... eventually more thought leads us to possible rules. So instead of thinking of them as possible rules, let's consider them as the rules and play with the rules and see what comes out...– MitchFeb 1, 2017 at 20:56