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This question is about the difference between a definition and an axiom.

However, it does not address the following point:

Whenever we define something, this is often written as a series of axioms. e.g.

Definition. A sigma algebra $\sigma$ over $X$ is a subset of $\mathcal P(X)$, s.t.

Axiom 1.

Axiom 2.

Axiom 3.

Moreover, I don't think we ever apply axioms without then defining a term as that which satisfies those axioms, unless my memory fails me.

But since objects, such as sigma algebra's, most often are characterized by multiple axioms, it makes sense to see those axioms as different things in and of themselves, so that we can talk about "the commutativity axiom" as a separate thing, and apply them to multiple definitions.

So does it make sense to think of axioms as building blocks of definitions?

Or are we losing something if we look at axioms and definitions like this?

user56834
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  • I think that once you have basic grasp on how modern mathematics work, this question answers itself. – Asaf Karagila Jan 15 '18 at 19:33
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    It doesn't matter either way. Look at them whatever way you find easier. Whatever way you choose, it won't make it either easier or harder to do the actual mathematics (calculate results, derive proofs etc.) –  Jan 15 '18 at 19:39
  • The axioms define implicitly some concept, i.e. a sigma algebra is exactly the "mathematical object" that satisfies all the relevant axioms. – Mauro ALLEGRANZA Jan 15 '18 at 19:50
  • Think of axioms as truths. Definitions are syntactic shorts cuts or equivalences. – copper.hat Jan 15 '18 at 20:00
  • Although the distinction may useful in informal discussions, in a formal system AFAICT, there is no need to distinguish between axioms and definitions. – Dan Christensen Jan 15 '18 at 22:09

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I don't really understand the question, in that I don't really understand what it would mean for the answer to the question to be "no," but it is common to think of certain axioms like commutativity and associativity as things which can be added or subtracted from other definitions, e.g. imposing commutativity on rings to get commutative rings, or removing the existence of additive inverses on rings to get semirings. Does that answer to your question?

Qiaochu Yuan
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  • I hope this will clarify my question: if the answer would no, an explanation could for example be: no, looking at axioms merely as building blocks does not take into account that axioms also have this other role X that is unrelated to definitions" or "no, because in meta-mathematics there is the diatinction between the two, and if we didnt make that distinction we would run into paradox X" or "no, because not recognizing that axioms and definitions are fundamentally different things because of X would lead you to conceptual error Y". – user56834 Jan 16 '18 at 06:04
  • Basically, if the answer is yes, then it seems to me that theoretically, we could drop the whole concept of an axiom, and simply think merely in terms of definitions and their logical implications. If the answer to my question is yes, then dropping the concept of axioms would merely be a practical nuiscance since for the definitions of every category that currently has the commutativity axiom, we wouldn't have a word (axiom) to refer to this commutativity property. But apart from this practical nuiscance it wouldn't cause any serious problems. – user56834 Jan 16 '18 at 06:09
  • @Programmer2134: I suppose you could drop the whole concept of an axiom, but I have no idea why you’d want to do so. It’s extremely convenient to be able to refer to axioms, e.g. when proving that something satisfies a definition. I’m still confused about why you would want to do this. – Qiaochu Yuan Jan 16 '18 at 06:16
  • Let me put the question in an even different way: if a really smart 12 year old came to you and said "I read on wikipedia that in mathematics there are such things as 'axioms'. I don't know anything about axioms, but I want to understand what theie role is in mathematics and philosophy". lets say that you wanted to give him a thorough answer. Would you explain to him that in mathematics, we have definitions of categories of mathematical objects (like $\sigma$-algebras) and that the axioms are the building blocks of those definitions? Or would you leave him with a fundamental misunderstanding? – user56834 Jan 16 '18 at 06:18
  • I am not actually suggesting to drop them.... I am trying to sharpen my conceptual/philosophical understanding of the role of axioms in mathematics. – user56834 Jan 16 '18 at 06:19
  • The idea of dropping them is merely a conceptual device, a thought experiment, to sharpen my conceptual understanding. – user56834 Jan 16 '18 at 06:20
  • Because let me stress that I dont think people USED to think about them this way. For example, Euclid thought of his postulates not as "parts of definitions", but as "universal truths", and it was not until modern mathematics came along that people started to study different categories based on mutually inconsistent axioms, without seeing them as somehow "universally true". If i'm not mistaken. – user56834 Jan 16 '18 at 06:31
  • @Programmer2134: the explanation I usually give is that axioms set out the rules of a game you'd like to play, like the games of chess (Hilbert was fond of a perspective along these lines as well). The game of group theory is to give examples of things satisfying the group axioms, and then to see what can be said about them. You're right that people didn't used to think about things this way; there are lots of interesting things that can be said about that. My impression is that people used to not have a clear understanding of the relationship between syntax and semantics. – Qiaochu Yuan Jan 16 '18 at 06:32
  • Anyway, I'm still not sure I really understand your question, although my answer is tentatively "yes." If you want to improve your conceptual understanding of mathematics my advice is mostly to just do more mathematics; philosophizing is fine and useful but it shouldn't take up more than maybe 30% of your mathematical time. – Qiaochu Yuan Jan 16 '18 at 06:34
  • I have asked another question here which maybe should clarify a bit what it is that I'm trying to understand: It is not the same question, but related to it: https://math.stackexchange.com/questions/2716510/when-to-write-something-in-terms-of-axioms – user56834 Mar 31 '18 at 19:31
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To use programming terminology, some "definitions" define interfaces and some define objects. The definition of $\sigma$-algebra that you've alluded to is a definition of an interface. For you to give me a $\sigma$-algebra means you are going to give me a set $X$, and collection of subsets of $X$, $\Sigma$, and a proof for each of the axioms.

On the other hand, I can define a specific $\sigma$-algebra. For example, the discrete $\sigma$-algebra on a set $X$ consists of the set $X$ and $\Sigma=\mathcal{P}(X)$. The proofs of the axioms are trivial. This is like defining an object implementing an interface. This is not an axiom or a theorem, though it involves theorems. This is often referred to as a "construction". (In a propositions-as-types context, "theorems" and "constructions" get identified, as a theorem is just a certain type of construction in that context.)

To directly answer your question, while axioms may be parts of definitions corresponding to "interfaces", those aren't the only definitions around and even in that case there are things other than axioms involved. For example, even in your definition of $\sigma$-algebra, you have the first line which is not an axiom. So I would say axioms can be called "building blocks" of "interface" definitions, they are not the sole "building blocks" of such. (If you really wanted to, in a set-theoretic context, I guess you could state that an "interface" definition is a collection of formulas with one free variable that will be taken as axioms. Your $\sigma$-algebra example would then have an additional axiom that $\exists X,\Sigma.Z=(X,\Sigma)\land \Sigma\subseteq\mathcal{P}(X)$ where $Z$ is the free variable. The other axioms would need to be modified to project out $X$ and $\Sigma$ from $Z$.)

If we look at mechanized proof assistants (many of which are based on the proposition-as-types principle), definitions in them usually involve a collection of parameters, a collection of new names each of which may or may not have a concrete definition (i.e. a [parameterized] construction) and which may require theorems to be proven of their own to be well-defined, and a collection of axioms involving the parameters and the newly introduced names. In proof assistants like Agda and Coq, defined notions induce definitional equalities that cannot be postulated as axioms and have a significant bearing what is or isn't provable.

Derek Elkins left SE
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  • I would have imagined that the discrete $\sigma$-algebra is the one generated by singletons, so the countable and co-countable sets... – Asaf Karagila Jan 16 '18 at 04:44
  • I took the name from Wikipedia, but as far as I can tell, this is also how Terence Tao uses it in his book on measure theory e.g. Example 2.3.3. He doesn't actually defined "discrete $\sigma$-algebra" though. He defines "discrete algebra" (Example 1.4.3) and states that it is a $\sigma$-algebra (Exercise 1.4.10). Example 2.3.3 does suggest that the $\sigma$-algebra structure on the discrete algebra is what he intends. I'm no expert on these, though. – Derek Elkins left SE Jan 16 '18 at 05:23
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The usual order of things is somewhat different from what you outline in your question. When one wishes to formalize a subfield of mathematics one (1) starts with a bunch of primitive notions that are left undefined, (2) proceeds to postulate certain axioms formalizing how they are interrelated, (3) gives definitions of further "compound" notions, (4) formulates theorems in the language thus developed, etc.

For example, in the field of geometry, one could use point and line as undefined primitive notions, then formulate axioms of, say, affine geometry (a unique line through a pair of distinct points, etc.), then define a further compound notion of a pencil of lines, etc.

Mikhail Katz
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  • But isn't it the case that your example of points and lines is outdated? My impression is that this is how Euclid's system was developed, but that in modern mathematics, we have in a sense dropped the concepts of point and lines as "primitive notions" and instead talk only about sets of objects (without saying anything about the objects themselves), and then list some formal statements (axioms) and state that e.g. "euclidean space" is a structure that satisfies those axioms. We have thus not really ever stated anything about "points". the definition of the space consists merely of its axioms – user56834 Jan 18 '18 at 14:16
  • If your basic reference is set theory then the only primitive notion you have is that of a "set" whereas everything else is a compound notion, including that of a line or a point. However, it is not obligatory to take set theory as your basic reference. – Mikhail Katz Jan 18 '18 at 14:58
  • Good point, you're right. About my main question, however. Maybe you are right in suggesting that the usual order of things is different from what my question suggested. However, when it comes to the role of axioms, as building blocks of definitions, it seems to me that your answer does not contradict this. In your answer, the axioms would still be the building blocks of definitions. In Euclid's case, his axioms are the building blocks of the definition of a "Euclidean space" (though saw them instead as universal truths). Different axioms would define e.g. a parabolic space. – user56834 Jan 18 '18 at 16:23
  • Let's take the definition of a group for example. Here we write down a few properties and declare them to be the defining properties of a group. Sometimes these are referred to as the axioms of a group. That's also a legitimate usage of the term. I am not sure what your question is though at this point. – Mikhail Katz Jan 18 '18 at 16:25