The Wikipedia article on “mathematical object” needs to be rewritten. It begins: “In mathematics any subject of mathematical research that can be expressed in terms of set theory is a mathematical object”. It also says, “Outside mathematics, a mathematical object is an abstract object that is referred to or occurs in mathematics”.
That is defining mathematical object in terms of a historical detail. Yes, it has been shown that most mathematical objects can be constructed from sets in one way or another, often in strange or unintuitive ways. However, that is a theorem, not a main property of mathematical objects. Besides, there are objects that exist in other categories but not in sets, such as models of untyped lambda calculus. Those categories involve proper classes of objects, not sets of them.
I would prefer some definition such as this: “A mathematical object is an abstract object defined by axioms”, together with explanations of abstract object and axiom. Mathematical objects should be distinguished from abstract objects such as “schedule” that change over time and also from objects in narrative fiction.
The article should describe the different points of view taken by philosophers and mathematicians who have written about the idea. It should refer to some of these articles and books:
Davis and Hersh, The Mathematical Experience (Mariner Books, 1999), sections on Mathematical Objects and Structures: Existence and on True Facts about Imaginary Objects.
Goodman’s article in New Directions in the Philosophy of Science, Princeton, 1998.
Hersh, What is Mathematics, Really?, Oxford University Press, 1997.
Stanford Encyclopedia of philosophy article on abstract object.
I have written about mathematical objects at length on abstractmath.org. It pulls together many of the ideas in the articles listed above.
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If you see something you think should be improved about Wikipedia, an appropriate response is to fix it yourself. It’s more likely to be effective than whinging in a blog.
I expect I will amend it myself, after I get comments on this post.
I should have added as reference: Gold and Simons, Proofs and Other Dilemmas, which was just published and which I have not seen yet.
“A mathematical object is an abstract object defined by axioms” is a problematic definition, as it blurs the line between theories and models, as it is understood (at least intuitively) by most contemporary mathematicians. Also, the definition is almost circular, as it explains (mathematical) object in terms of (abstract) object.
For example, the concept of a “group” is defined by axioms, and therefore would fit your definition, yet most people would not call “group”, as a concept, a mathematical object – rather, individual groups are mathematical objects.
Perhaps you feel that “abstract object” already excludes things that are merely concepts, but that presupposes a definition of “abstract object”, and seemingly becomes circular.
(You could of course argue that the concept of a “group” is embodied in the category of groups, which is itself a mathematical object – but this wouldn’t be right, because the category of groups itself is not described by axioms.
If one believed your definition, one might as well define an object to *be* a set of axioms, without any consideration of the question whether the axioms are consistent (existence) or complete (uniqueness).
On the other hand, some things that are clearly mathematical objects, such as the finite group given by the following multiplication table (insert table here), are not typically described by axioms, but are given by a construction inside some fixed universe (which is often, but not always, a universe of Sets).
Actually, models of the untyped lambda calculus can be quite conveniently expressed in terms of set theory. Only a category theorist would jump to the conclusion that a set equipped with additional structure can no longer be called a set (or equivalently, that a set can’t have any structure besides cardinality).
I am certainly not advocating that set theory is the only possible foundation for mathematics. But surely it is one of several possible equivalent foundations, so that the original Wikipedia definition is correct. It only stipulates that the object “can” be expressed in terms of set theory, not that it has to be expressed that way. So the use of set theory in that definition is without loss of generality.
What is an example of a mathematical object that cannot be expressed in terms of set theory? Something that is a proper class doesn’t qualify in my opinion, because it can be expressed e.g. in Goedel-Bernays set theory (and possible in ordinary set theory as well, modulo some admittedly ugly coding).
Your definition sounds to me like you’re defining abstract mathematical objects. For example, any particular model of a theory should count as a mathematical object, as much as the theory itself, shouldn’t it?
So, maybe the definition could go “built from a set of axioms using standard logical constructions”? Not quite satisfactory yet.
According to the page history, the page has only existed for a week. I suspect there is no consensus as to what a “mathematical object” is, in general.
There are no references on the page. Is there a proper definition of “mathematical object”? The page shouldn’t end up as “original research”. Would it be best to just delete the article?