Are there mathematical objects that have been proved to exist but cannot be described in words?

Question

This might be a very stupid, and possibly philosophical question, but attempt to apply mathematics to everything plus inspired by this question caused me to ask this question

1. Is there any mathematical object that has been proved to exist but cannot be described in words?
2. If the answer is yes, are these things in general meaningful to study in mathematics (that is, can be applied to prove some theorems or demonstrate some additional properties of some mathematical objects (or even a possible real life application)?

The closest example I have found so far to question 1 is the Hamel basis in $\mathbb{R}^\infty$ which via Zorn's Lemma is shown to be exist but no known way to actually construct it explicitly. But we still able to describe some of its properties, such as

1. It is uncountable

2. It is a basis of $\mathbb{R}^\infty$

There are two reasons that motivates me to ask this question

1. Any mathematical objects I have read so far, no matter how abstract, can still be described by some sentence and definition statements that follows, or at least written as a relation between two or more mathematical objects. For example:

In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "size". The mathematical existence of such sets is construed to shed light on the notions of length, area and volume in formal set theory.

Thus I am wondering if there exist counterexamples that cannot even be described in words.

2. Any two life experience cannot be related to each other in general by some mathematical relations. For example in real life "the experience of seeing the color red" is often specific to different people but there is no way to tell how it is different. But experience is vague and not a mathematical object. Thus I am interested in a mathematical 'analogue' of experience.

=================================================================

Edit: Having read this link as suggested by the comments and answers (although I am not sure if I fully understood it), I guess one way to define "describable in words" can be phrased as the following

Consider some mathematical objects $A$, $B$, $C$, etc., not necessary countable, living in some mathematical object $\mathbf{M}$ (More general than a category or a set). There are relations $\phi_\lambda$ where $\lambda \in G$ ($G$ is a mathematical object, not necessary countable) that given something $m$ in $\mathbf{M}$ one can say for example:

$$\phi_1(m)\text{ relates m to A, B}$$ $$\phi_x(m)\text{ relates m to A}$$ $$\phi_{\delta}(m)\text{ relates m to {A,B}}$$ $$\phi_{\omega_0}(m)\text{ relates m to $\emptyset$}$$ etc.

where the relation does not necessary generate a unique output or a set of outputs nor structure preserving for each mathematical object, thus not a morphism in general.

An identity relation can be defined as follows (there are many identity relations because $\mathbf{M}$ is not necessary a group) :

\begin{equation} \mathbf{id}_0(\cdot) \text{ satisfies "has the same property as $\cdot$"} \end{equation}

A concrete example:

$\mathbf{M}$ is the field of complex numbers $\mathbb{C}$, with

$\phi_\lambda$ are some properties of complex numbers. e.g.

$$\phi_1(\cdot) \text{ is $\{\cdot \text{ such that } i\cdot-1=0 \}$}$$ $$\phi_2(\cdot) \text{ is $\{\cdot \text{ such that } i\cdot \}$}$$ $$\phi_3(\cdot) \text{ is $\{a \text{ such that } a^2=\cdot \}$}$$ $$\phi_4(\cdot) \text{ is $\{\cdot \text{ such that } i\cdot \}$}$$

Then consider an element $z\in \mathbb{C}$

$$\phi_3(z)=\sqrt{z}$$ $$\phi_1(z)=-i$$ $$\phi_2 \circ \phi_1(z)=1$$

etc.

Another example:

$\mathbf{M}$ is a set of objects. Then suppose

$$\phi_o(\cdot) \text{ $\cdot$ satisfy "is vacuously true for a", where a satisfy the property "for all ..."}$$

Then

$$\phi_o(m)=\emptyset\text{ or }\text{"$\left\{\mathbf{v}_1\right\}$ is an orthogonal set for all $\mathbf{v_1}\neq\mathbf{0}$"}$$

So my question 1 boils down to:

Any example of at least one mathematical object $q$ in $M$ such that the only relation (as defined above) that exists for it is the identity relation as defined above (that is, cannot be described in terms of other mathematical objects except by the statement "has the same properties as itself"?), or put in terms of maths

$q$ are objects such that $$\phi_\lambda(q)=q$$ implies $$\phi_\lambda \text{ is } \mathbf{id_0}$$

Answer 1

You'd have to be more precise about what you mean by "describable in words". But you could argue that some irrational numbers can be described in words, like $\pi$ is the ratio of the circumference to the diameter of a circle. So if you consider each real number a mathematical object, then we could never possibly describe all of them because the set of things we can describe in language is necessarily countable, there are only a countable number of ways to assemble letters into words into sentences. But there are uncountably many real numbers. So inevitably most of them could never be described.

The reals are describable as a set, but you cannot describe each and every one of them in a way that distinguishes their individuality. Incidentally we can describe each and every element of $\mathbb N$ individually, because every natural number can be represented by a unique finite sequence of characters in the set $\{0,1,\dots,9\}$. The same cannot be said of $\mathbb R$.

Answer 2

Anything coming from the axiom of choice really. Such as the well ordering of the reals. Is there a known well ordering of the reals?

Answer 3

We can do it using set theory. The number of definable objects is countable, but the number of things that exist is uncountable. So something exists which isn't definable.

[edit]

Interestingly, you can't express definability in ZFC, in ZFC. If you try to, you get the following contradiction: https://mathoverflow.net/a/204794/1682.

Answer 4

To give another example from non-standard analysis (i.e. analysis with infinitesimals):

In non-standard analysis you extend the set of reals in a similar way you extend $\mathbb Q$ to $\mathbb R$ with Cauchy sequences:

1. You take the set of all real sequences.
2. You define a equivalence relation on the set of sequences. Two sequences $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ are equivalent if $x_n=y_n$ "for almost all $n\in\mathbb N$".

To define the concept "for almost all $n\in\mathbb N$" so called ultra filter are used, which existence can be shown with the axiom of choice. Until now no constructable proof is known. So you can show that

• The equivalence class of $(\tfrac 1n)_{n\in\mathbb N}$ will be an infinitesimal, i.e. a positive nonzero number smaller than each $q\in\mathbb Q^{+}$.
• The equivalence class of $(n)_{n\in\mathbb N}$ will be infinitely big.

But you will not know, whether the equivalence class of the sequence $(0,1,0,1,0,1,0,1,\ldots)$ represents 0 or 1.

(Note: What $(0,1,0,1,0,1,0,1,\ldots)$ is, depends on the taken ultra filter. There are some, where this sequence represents 0 and some where the sequence represents 1).

Answer 5

Things that are not describable in words exist as disturbances, obstacles to our understanding of the world, until we make them describable, i.e. until we find a conceptual metaphor rich enough to bring them to existence in language, as well as in conception. This is the adventure of science. In the history of mathematics this is particularly evident. There was once a time when irrational numbers weren't "describable in words". Now they are taught to children.

Are there objects that can keep resisting our attempts to tame them? I think there are, and some of the other answers tried to explore some of these "occurrences". More than once a mathematician was able to demonstrate things s/he couldn't convince him/herself of. It can be argued that these objects are part of our ordinary experience of the world, and we just don't mind their presence. This ineffable dimension of being is what some philosophers call the Event, or the Real.

Some say that modern science has recently found unsurpassed limits in its ability to digest the abnormalities it encounters, limits it can no longer expect to negotiate. And yet, it keeps describing new things. In words, of course. And also designating them by symbols, or creating images of them.

Will there always be objects that are indescribable in words, waiting to be "domesticated"? Now, that is a very interesting philosophical problem.