Does math admit (care about) the existence of an entity?

Question

Philosophers always concern about the existence of entities. For example, they discuss the propositions like "The bird exists." or "Unicorn does not exist". But in math, existential propositions are not formulated to "∃ entity" or "entity ∃" but something like "∃x HasUnicorn(x)", i.e., the "exists"("∃") is always followed by a variable which is nothing but a placeholder. The semantics of the existential formula are to substitute a specific object in the discourse of discussion for the variable and consider its truthness. But mathematicians seem to never concern about the existence of that specific object, they just concern if that object has some property. So, how is the existence of specific entity represented in math?

Answer 1

Within a domain of discourse, yes, very much so. But in the same way that "Unicorns exist" could be rephrased as "There exists an animal that has unicorn properties". So I don't admit the difference you draw between "Enity exists" and "Exists x HasProperties(x)".

So within the theory of numbers, the questions: "Do prime numbers exist?", "Do odd perfect numbers exist?", "Do square prime numbers exist?" are all reasonable mathematical statements and questions of existence.

But when mathematicians ask if numbers exist, external to the theory of numbers, then they are acting as philosophers. Some mathematicians do ask such questions, but when they do so, they aren't doing maths. Just as some mathematicians run marathons, but this doesn't make marathon running part of maths!

A biologist might ask "do unicorns exist?" as a biological question, meaning "Do animals with the properties of the unicorn exist?" For the biologist, the meaning of "exist" is not a significant part of the question of the existence of unicorns. On the other hand, if they are asking the ontological question of existence, they are not doing biology. The analogy with mathematicians is sound. If I ask "do odd perfect numbers exist?" I can ask that as a mathematician, in which the meaning of "exist" is well defined within number theory, or as a philosopher, in which the meaning of "exist" is moot.

Mathematical processes: logical reasoning from axioms, for example, can't establish the existence of entities outside of those axioms. So the question "do numbers exist" is a philosophical one, not a mathematical one.

Answer 2

You ask:

Does math admit(care about) the existence of an entity?

If you consider math a formal science, and the question of existence an ontological and therefore philosophical question, then its best to say that contemporary, working mathematicians tend not to care so much about what it means for abstract objects (SEP) to exist, and leave such questions to philosophers of mathematics, preferring instead to perform the existential quantification with symbols and leave the truth of the matter intuition (SEP). Historically, there was little distinction between philosophy and mathematics with the ancient Greeks like Pythagoras and Plato being both philosophers and mathematicians. But today, working mathematicians are interested in the science of math, whereas philosophers, particularly philosophers of mathematics, are more interested in the philosophical question.

Andrew Wiles is an example of a working mathematician, and Stewart Shapiro is an example of a philosopher of mathematics. Wiles is famous for his solution to Fermat's Last Theorem, and immersed himself for decades in the mathematical and logical sciences to find a proof related to the conjecture. On the other hand, Shapiro is famous for his philosophical theory which is about various questions and is called structuralism. Wiles dealt with language and formal systems that were distinctly mathematical and logical in character, whereas Shapiro's work deals with natural language that seeks to describe and explain and categorize the mathematical logic of working mathematicians. Where the two men might share knowledge is in the domain of discourse known as mathematical explanation (SEP).

A working mathematician looks for proof and provides mathematical explanation, whereas a philosopher of mathematics seeks to put the practice of mathematical explanation into a philosophical framework to explain how explanation works. The former might be understood as the language game that mathematicians play, and the latter as the language game philosophers play. Mathematicians do math, whereas philosophers are more interested in understanding in a greater context what exactly it is that mathematicians do and how it relates to other topics like science, language, and the mind.

Working mathematicians generally agree on axioms and syntax when exchanging ideas and focus on proof theory, where as philosophers are quite disputatious and are often looking to answer metaphysical, epistemological, or ontological questions that have little to do with mathematical proof. One of the fascinating debates in philosophy of logic and mathematics is whether or not mathematical structure is real or not. Another is whether it is objective and mind-independent. Frege was an antipsychologist and claimed that mathematics and logic were not dependent on the mind. Brouwer was an intuitionist (SEP) who disagreed and sent some working mathematicians down the path of mathematical constructivism as a technique in math. Plato argued there was an alternative realm. Shapiro argues that structures are real and objective. Lakoff argues math is nothing more than neural computation. A good, brief introduction to the philosophy of math is Philosophy of Mathematics (GB) by Linnebo.

So, does math care? Mathematics is a theory, and doesn't care, but the people who do it can roughly divided into two groups: those who don't, the working mathematicians who tend towards Platonism and realism based on their intuitions, and those who do care, mathematical philosophers who have a bewildering array of views about mathematical existence of entities. In this way, formal scientists are a lot like physical scientists. They often see such philosophical disputes as irrelevant and don't spend much time on the question. They simply declare in their papers and books things exist and move on.

Answer 3

Reformulating your question a little:

Do mathematicians care about the existence of (mathematical) objects?

Yes, they do. Very much so. But they leave it up to you (anyone) to specify what it may mean to say that something exists (as long as you are consistent in your use of it).

You wrote:

The semantics of the existential formula are to substitute a specific object in the discourse of discussion for the variable and consider its truthness. But mathematicians seem to never concern about the existence of that specific object, they just concern if that object has some property.

That statement is, I believe, either not true or misleading.

There are essentially two contexts in which existence statements are made in mathematics, that is, statements that say "there is ... " or "there are ..." or "there are no ...". The more common context is the context of mathematical theorems that apply to some given set of objects. The other context is the context of axioms that are intended to establish what we want to consider as given.

An example of the first is Euclid's proof that there are infinitely many prime numbers. This can be read as: Given any finite set of primes, there is is a prime that is not in that set. We can prove this by specifying how to construct a new prime from the given primes. So, "there is ..." in this case implies "we can find a new one, we can construct it". It's tantamount to saying, "I have a method by which, for any finite set you throw at me, I can make a particular new prime that is not in the set (and that method is: multiply all the given primes with each other and add one)". (Note how this statement has the pragmatic setting of a dialog, a game of challenge and response.)

If you accept LEM (the Law of the Excluded Middle), then non-constructive proofs are also allowed. In that case, it seems to become a bit vaguer what "there is ..." means, since the proof may not give a method to find a particular object satisfying the stated property. One reason why intuitionists disallow use of LEM is precisely that this makes it vaguer (too vague) what it would mean to say "there is...".

In the context of axioms, an axiom may be an unqualified existence statement, a statement not restricted to any domain of discourse or to any "world". For example in classical set theory (ZFC) we have

There is an empty set

We need this axiom to "get going". Without it, all other axioms and theorems would become purely conditional (on any sets existing). With it, we can prove (based on the definition of "empty" and on the extensionality axiom) that there is only one empty set. This justifies referring to the empty set. In other words, in the light of this unqualified existence axiom, we justify talk of "the empty set" as a particular object. And with that (plus the other axioms) we can start proving existence theorems.

The justification of the axioms can itself only be informal. (You may be able to set up a slightly different system in which a former axiom becomes a theorem based on other axioms, but then those new axioms are either not justified or justified informally.) If we can at all (informally) conceive of sets, collections of things, then it seems there is no problem conceiving of an empty set. The simple physical analogy is having a box filled with items. If we can imagine that, then we can also imagine having an empty box. The basic justification is logical consistency -- which may be just an unproven or even unprovable assumption. In a way, this seems to imply that axiomatic existence statements are mostly a convenience, only constrained by the requirement that no logical contradictions should be provable. They are more than just a convenience, however, in sofar as they define the intended ontology of a system: they establish the specific universe of discourse itself.

Answer 4

A Philosopher and a Mathematician are out at night. The Philosopher says to the Mathematician "Suppose we each have a beer." They walk into a bar. The Bartender says "Would you each like a beer?" The Philsopher says "Yes, please." The Mathematician says "No thanks, I've already got one."

The Mathematical notion of "existence" is a hypothetical one. One does not need to actually possess physical quantities of beers to reason about what will happen when we put quantities of beers together, or take them away, or divide them up, or whatever. This hypothetical property is easy to miss, because it is generally buried deep deep deep down in the system's axioms.

You seem to be familiar with the phiosophical notion of existence, so I will leave this topic alone, and focus on how it relates to Mathematics.

The important feature to note right up front is that Mathematics concerns itself with formal systems. While these systems are generally designed to model reality, they are not real. Here, by "model reality", we mean that the mathematical reasoning is all in the abstract and hypothetical - but that if one were able to instantiate the abstractions with "real" objects and if those objects fufill the initial hypotheticals - then the conclusions of the mathematical reasoning, as applied to the real objects, will all hold in reality. This is a very cool feature, because it means that we don't actually need to have physical beers before we can start our beer-accounting logic. For example, we can design beer-addition that will help in our real-world beer problems before we even have any physical beers to deal with. Our reasoning on beer-addition will be useful night after night, and we need not re-reason the topic for each physical manifestation of beers.

Historically, this distinction was not made, and, indeed, philosophers such as Plato claim that mathematical entities are real. However, the position that mathematical abstractions are real is not common in the modern Mathematics community. Since all of the subjects of mathematical statements are merely symbolic entities in a formal system, then obviously any talk of "existence" must be construed within this context. The context is that the reasoning is all in the abstract and hypothetical: We need not have actual "existing" beers to reason with - the reasoning is independent of the real-world objects.

To illustrate, we can reason hypothetically about what will happen when we put 2 beers and 3 beers together without actually having beers. We can abstract this to situations of oranges, or bicycles, etc. While a quantity of beers may exist or not exist in the philosophical sense, a hypothetical quantity of beers definitely does not exist in the philosophical sense. However, we can still talk about our hypothetical beers by simply hypothesizing them: "Suppose that you have two beers and I have three beers." We can now talk about whether "there exists" (in the hypothetical sense) a quantity of beers that occurs when you and I hypothetically put our respective hypothetical quantities of beers together.

As mentioned above, the hypothetical nature of the discussion is usually buried deep down, and is therefore implicit in the context of doing Math at all.

Answer 5

Your question concerns the existence of specific objects vs. defining objects by their properties. This is one of the main questions in foundations of mathematics.

Penelope Maddy gives a detailed discussion of the pros and cons of the various approaches (history, practical aspects, coverage), Penelope Maddy - What do we want a foundation to do? Comparing set-theoretic, category-theoretic, and univalent approaches, which is a relatively approachable with some background in set theory and logic.

The traditional (19th-early 20th century) answer was to use base objects (the sets of Set theory) and build up on them. Many properties (though not all) defined proper (sub)sets of the "universe" of sets. That allowed ignoring the base sets and arguing with the properties as axioms. (Other properties lead to paradoxes, so certain properties or their application were ruled out later on.)

But there were other systems developed, namely Category theory that began with very abstract structures, unspecified "objects" and "morphisms" (mappings) between them, without necessarily defining base entities (the objects could be, for example, sets, groups but also individual numbers or any other entity that fit the schema without the need to specify the elements of these objects). This proved incredibly useful in many applications but hasn't, as of today, approached the universal "coverage" of set theory (see paper below).

A third option is Homotopy Type Theory, which is formally rather similar to computer syntax (partially what makes it attractive - statements and proofs can be checked or even generated automatically).

So the general picture seems to be: either you start at the "bottom", defining some base elements and working upwards towards various properties, or at the "top" with properties and their relations and proceed downwards to reach familiar objects (or a hybrid of the two).

Answer 6

"So, how is the existence of specific entity represented in math?"

How are we poor mathematicians supposed to know which entity is having its existence asserted? How can we recognize it, how can we specify it? The best approach we've found is to describe the entity by the properties it satisfies, so we use ∃xP(x).

Do you have some alternate, possibly better way of specifying entities? Let's hear it!

Answer 7

A typical example for the existence resp. non-existence of an entity with a specific property is the solution of arithmetic equations:

1. If the underlying domain is the set of real numbers then

   ∃x: x^2 = +1

   not ∃x: x^2 = -1

2. If the underlying domain is the set of complex numbers then

   ∃x: x^2 = +1

   ∃x: x^2 = -1

Hence the existence of entities with a certain property is always relatively to a given domain of elements.

Answer 8

OK, I see several interpretations of, or answers to your question, depending on what you mean by "existence".

• "Do the mathematical entities exist in a physical sense?" - no. Non-applied maths is perfectly happy to work in its own world of axioms and purely mathematical objects which exist purely as concept or idea, and in no way, shape or form physically. Large swaths of maths are built on purely invented and somewhat arbitrary axioms that do not require or speak about anything that could (or needs to) exist in a physical sense.
• "Does a function or other mathematical object exist that fulfills these criteria?" - This would depend. For example I had a course on Abelian Groups at uni, and it was fully self-contained. I.e., we started from axioms specific to this course, and ended with a large amount of theorems to work with these axioms, to talk about questions involving Abelian Groups. I cannot remember even once working on a concrete example in the course or the self studies. But of course many questions were like "Assume a group exists with these attributes" or "Prove that an element exists which ...".
• On the other hand, sometimes mathematicians are of course interested to actually find some mathematical object that is supposed to exist according to some theorem. In these cases, they are very much interested in the existence (even if only the existence of something representing a concept or idealistic thing, like a function over R). That's were you stray more into more pragmatic (but possibly still not "applied") mathematics; i.e., ways and rules or best practices to actually solve a problem.
• Finally, in my time and at my uni, Computer Science was considered a branch of Mathematics. In Computer Science, we have a lot of structure around existence - i.e., things like lambda calculus and other formalisms of predicate logic, constraint logic etc.; often even more abstract or powerful or multidimensional than other parts of maths. Here, we quickly become "applied" as we start writing actual algorithms and programs, and as soon as you do that, you will run your programs with actual inputs and outputs. Even if these are still playthings (or "libraries" which do not actually do anything practical), existence is quite the thing to bother about, even if it (whatever "it" is) only exists in the form of electrons in RAM. Even more so than in other applied branches of maths (i.e. physics), the borders are very much fuzzy there.

Answer 9

There is no difference between the existential operator in math and normal, everyday existence statements other than the formalization of math. "There are mammals that don't bear live young", for example, would be represented mathematically as ∃xM(x) where M represents the property "is a mammal that bears live young". Statements like "Ghosts exist" can be written formally as ∃xG(x) where G represents the property of being a ghost.

These propositions just mean "there are things that satisfy some property" which is no different than what is meant in mathematics by ∃xP(x) where P represents "is an even, prime number" or "is a square root of 2".

There is a philosophical use of the word "exists" that is different from existential quantification, but that isn't the normal-language use. The philosophical use involves the metaphysical notion that existence is a quality that some things have. One might for example, list some qualities of the moon as "has a rocky surface, has no air, has existence, etc." But the qualities of Endor (a moon in Star Wars) would have the list, "has a rocky surface, has air, has no existence, etc." The list for Endor is just a fictional list to clarify the idea; most philosophers who think existence is a quality would say that there is no Endor so Endor has no qualities at all; that before anything can have any qualities, it must have existence as a foundational quality (that's why I used "quality" instead of "property"; existence can't be a property if it is a prerequisite for having any properties). Not all philosophers accept this idea of existence as a quality.

The mathematical and normal-language concept of existence can be construed as a second-order property of properties. In other words, ∃xP(x) can be construed as meaning E(P) where E a property of properties that mean a property has instances. This was how Frege, the inventor of quantifiers, described existence.