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

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    $\begingroup$ See en.wikipedia.org/wiki/Computable_number and math.stackexchange.com/questions/462790/… $\endgroup$ – Chappers May 16 '15 at 16:25
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    $\begingroup$ What do you mean with 'described in words'? $\endgroup$ – Git Gud May 16 '15 at 16:27
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    $\begingroup$ PS I really wish there's an option to accept more than one answer. This is like the 3rd question without an accepted answer since there is more than one which works... $\endgroup$ – Secret May 16 '15 at 16:44
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    $\begingroup$ If you can't describe it in words, how are you referring to it when you prove that it exists? $\endgroup$ – Qiaochu Yuan May 16 '15 at 17:40
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    $\begingroup$ Every time questions like this come up, people immediately diagonalize over sentences of the language to conclude there are countably many definable objects, and it's impossible to get them to stop. If ZFC is consistent, then there must be models of ZFC where everything in the model actually is definable, even though it seems like there should only be countably many definable objects. Please read Joel David Hamkins's more detailed answer on MathOverflow. $\endgroup$ – user2357112 May 17 '15 at 2:04
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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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    $\begingroup$ +1 for it is not clear what's meant with "describable in words"; $\endgroup$ – Stephan Kulla May 16 '15 at 16:31
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    $\begingroup$ The reals are describable as a set, but you cannot describe each and every one of them in a way that distinguishes their individuality. $\endgroup$ – Gregory Grant May 16 '15 at 16:35
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    $\begingroup$ You can describe the individual real numbers with infinitely long sentences. $\endgroup$ – Matt Samuel May 16 '15 at 19:01
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    $\begingroup$ Um, simply list the digits. One word for each digit. Why should I have to finish? It's an infinitely long sentence. Taken as a whole it describes all of the digits of pi. Note I never claimed this was practical or anything. $\endgroup$ – Matt Samuel May 16 '15 at 19:16
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    $\begingroup$ We can't say that there are countably many definable objects, because we can't actually construct the correspondence between a definition and the object it defines. If we could, we could do things like define the lowest undefinable ordinal. See Joel David Hamkins's more detailed answer on MathOverflow. $\endgroup$ – user2357112 May 17 '15 at 1:41
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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?

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  • $\begingroup$ Of course, we've also proved that "there is a well-ordering of the reals" is independent of ZF - which, to some degree, means, "we can't actually describe a well-ordering, so we took the existence of one as an axiom, because it suits us even though we can't prove it from other principles." $\endgroup$ – Milo Brandt May 17 '15 at 4:37
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    $\begingroup$ It is consistent with $\mathsf{ZFC}$ that there is a definable well-ordering of the reals. For example, it follows from $\mathsf{ZFC} + V = L$ (which is consistent relative to $\mathsf{ZFC}$.) $\endgroup$ – Trevor Wilson May 17 '15 at 20:12
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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.

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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.

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  • $\begingroup$ What do you mean by "definable"? The real numbers, which are uncountable, are considered a definable set. $\endgroup$ – chepner May 16 '15 at 18:22
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    $\begingroup$ @chepner I believe in the case of the real numbers, "definable" would refer to the fact that the set itself is defined by a formula, not that every element in that set is definable (which is not the case for the real numbers, since they are uncountable). So there are real numbers that can't be defined, even if the set containing all real numbers can be. $\endgroup$ – Frxstrem May 16 '15 at 18:47
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    $\begingroup$ @chepner There is a real number which is not defined in the language of ZFC, in the sense that there is no formula which describes only that object. $\endgroup$ – man and laptop May 16 '15 at 19:27
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    $\begingroup$ This answer is not quite right: see mathoverflow.net/a/44129/1682 or mathoverflow.net/a/204794/1682 $\endgroup$ – Trevor Wilson May 17 '15 at 5:15
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    $\begingroup$ @TrevorWilson Very interesting. That means you need something that can express "definable in ZFC", and ZFC itself clearly can't without being inconsistent. This is worth an edit. $\endgroup$ – man and laptop May 17 '15 at 11:10
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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).

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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.

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    $\begingroup$ This post reminds me of a joke: Dean, to the physics department. "Why do I always have to give you guys so much money, for laboratories and expensive equipment and stuff. Why couldn't you be like the math department - all they need is money for pencils, paper and waste-paper baskets. Or even better, like the philosophy department. All they need are pencils and paper." $\endgroup$ – Benjamin Lindqvist May 17 '15 at 9:30

protected by user21820 Nov 12 '18 at 6:07

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