Soft Question – Mathematical Objects Proved to Exist but Cannot Be Described in Words

soft-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.

  1. 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.

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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
$$\phi_\lambda \text{ is } \mathbf{id_0}$$

Best Answer

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 $\mathbb R$.

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