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If a sentence is never written, spoken, or even thought of, does it still exist? Let me illustrate what I mean. Suppose there is a mathematical sentence, say in some first-order language L, which is a googolplex symbols long. Since such a statement can never be written down in the lifetime or volume of the observable universe, does it make sense to say that it still exists? I suppose we would need a definition of "sentence" to answer this question. Maybe there are two definitions here, one of them being the concrete physical inscription, and the other being simply an abstract sequence of words. Anyway, what have philosophers written about this? I would like to see some references.

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    What does it mean for you that something exists?
    – Johan
    Apr 4 at 22:52
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    Godel sentence was not written, spoken or even thought of obviously before Godel... Apr 4 at 23:17
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    "Exist" is a very vague word that does not mean anything in particular. All sentences "exist" in the sense of abstract objects, for example. And they may not "exist" as concrete physical inscriptions even if the time it takes to inscribe them is very short. Or even if someone once inscribed them but the inscription deteriorated.
    – Conifold
    Apr 5 at 7:27
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    You are thinking of that sentence right now, aren't you?
    – Tvde1
    Apr 5 at 9:10
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    @Tvde1 is the sentence in the room with us right now?
    – Brondahl
    Apr 5 at 12:21

10 Answers 10

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In math and logic structures can be said to "exist" even if the current universe may not hold an instance of them. Like "for every prime number there exists a larger prime number". This does not imply physical existence, just existence within an abstract set, as opposed to no Element within a set satisfying such a condition.

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  • I'm not downvoting, but your claim is only true if you choose to ignore Brouwer and constructive mathematics. plato.stanford.edu/entries/mathematics-constructive
    – J D
    Apr 5 at 15:02
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    Constructive maths is fine with this sort of conceptualization. In particular, if an algorithm can be given for a construction, then this implies existence of the constructed object; this is known as realizability. I think @JD is thinking of ultrafinitism, whose adherents often deny the existence of abstracta which can't be physically instantiated.
    – Corbin
    Apr 5 at 17:02
  • I don't agree with this analogy. In the math description you gave, you technically need to add "in the set of all real numbers". You have to first define the set of possible outcomes. Likewise, if you declare the set of sentences as "all possible combinations of letters and spaces", then yes, you can claim that all sentences exist, whether we know them or not. Apr 5 at 21:59
  • The question as stated is not about infinites, and the size constraints seem speculative. In terms of philosophy, I still think the question revolves around the word "exist", and the answer that in math an logic that typically implies "in an abstract set" fits the question. There may be other similar questions though to which other answers might be better.
    – tkruse
    Apr 8 at 11:38
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ultrafinitists will say the googolplex-long sentence does not exist, precisely because the concrete physical inscription cannot exist, while most everyone else will agree it does 'exist in principle'/'exists abstractly' because it does exist as an object some weak theory of arithmetic as, say, PRA

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    That's more than a stretch to say that almost everyone's ontological compass is some weak theory of arithmetic.
    – Johan
    Apr 4 at 22:59
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    @Johan most everyone who's ever given a thought about formal sentences/grammar in logic and mathematics is happy to use and consider as 'real' stronger theories and their objects, I don't really understand your point
    – ac15
    Apr 4 at 23:05
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Your title includes "never been thought of" while your actual question body clearly addresses things we can think of (the very long sentence).

Let's first state that one of our great capabilities is that we can abstract. We can count items — Imelda Marcos had 6000 shoes. We can line them up and behold them. We can abstract from the shoes though and mentally marvel at this astonishing number, 6000. Does this number exist? Most people would agree. We can abstract further and put all the natural numbers in a set. Common use of the word "exists" among people who have nothing better to do than think about such idle concepts would indicate that yes, the set of natural number exists, including every single number in it ("How many push-ups can Chuck Norris do?" — "All of them!") even though this set does not have a concrete physical representation in a finite universe, let alone in our even more finite minds.

This shows that we can hold abstractions in our minds. Your long mathematical sentence surely exists in this abstract sense — it is not even infinite! To an individual this existence is no lesser than that of Imelda Marcos's shoes because actually all we ever manipulate mentally is a model of the world existing in our minds. Made-up things in this model world are of no lesser significance or existence to the individual than representations of exterior things we have perceived through our senses. Whatever model we hold in our minds is our "reality". This is obvious in paranoid people. The difference to "sane" people is that they usually cannot convince anybody of their ideas; the reality they live in is very much their own.

We see that a shared reality depends on whether we can communicate the things in our minds properly. Everybody who accepts the communicated information will agree on that reality, sometimes with tragic consequences (tragic to people who rejected that communication); for example if somebody convinces others of their suicide death cult.


The question whether things nobody ever thought of exist is even more intriguing. Is mathematics a creative endeavor ("inventing" things that didn't exist before) — or is it merely a process of discovering existing laws? I'm in no position to answer that question and simply point you to the google search "mathematics invented or discovered". But there is a certain similarity to Michelangelo's quip about the statue always being there in the marble block; his job, he said, is just to chisel away the marble obscuring it.

Because that humble and witty statement is obvious nonsense I tend to think that mathematics is a creative endeavor, like writing poetry. After all, one could argue that all combinations of letters — including all past and future poems — already "exist" (in the sense that all natural numbers exist). The poet's job then would simply be to make a good choice from that pre-existing collection. That is pretty much as nonsensical as Michelangelo's quip (sorry, Mick).

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No-one has mentioned types and tokens.

Only some sentence tokens exist (even if random fluctuations in matter can be tokens of sentences, I'd have thought that the physical world does not exemplify an infinte number of structures and does not include an infinte number of shapes or sounds, and so not every possible instance of language occurs, as that is infinte, as that is infinite), and these are occurences of types

' I wondered lonely as a cloud. I wondered lonely as a cloud.

Two tokens of one type of sentence.

Someone might suppose types can exist withot occurring in a token, but it seems ontologically flabby to suppose that every token (of a sentence) that can occur, is metaphysically possible, would be an occurrence of a type that actually exists.


Modal realists might think that everything possible exists.

I suppose that abstract mathematical entities might exist and these could be translated into linguistic sentences. I wondered lonely as a cloud:

0100100100100000011101110110111101101110011001000110010101110010011001010110010000100000011011000110111101101110011001010110110001111001001000000110000101110011001000000110000100100000011000110110110001101111011101010110010000101110

If intereptation defines what some language is, then maybe every sentence exists as they can all be read into anything.

etc.. It probably depends on what can be a 'sentence' and your ontology in general (what if nothing exists?)

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    kinda bullshitty, but you get what you paid for.
    – andrós
    Apr 6 at 11:33
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The best philosophical definition of existence I've found is based on the interaction subject/object:

An object exists for a subject if the subject can interact with it in some way.

So, existence is subjective (obvious: God exists for some, not for all), and it is not necessarily physical, or conceptual.

For example, if such sentence is named X, it exists (for me) as such, because I can interact logically with it (for example, I can put it in the list of things I can't observe). You judge if you can interact with it, depending on your constraints.

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  • Is non-interaction also a form of interaction (like the empty set is a form of set)? I would really like to interact with sentences that do not interact with me! Apr 5 at 8:34
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    this definition seems to be self-defeating for abstract things. They would always exist in a way, because any subject can interact with them, even if the only meaningful "interaction" is refuting their existence. The example of God illustrates that quite well - the abstract notions of Jehova (any of the interpretations), Zeus and the Flying Spaghetti Monster can be interacted with by everyone, but they are not expected to exist physically in the world at the same time. This definition has no counter examples.
    – Chieron
    Apr 5 at 8:54
  • @Peter-ReinstateMonica all reality happens in your head, where there is you, and there are objects. Language predicates always express such interaction: Peter [subject] likes interacting [interaction (also, verb)] with sentences that... [object]. When you drink water, there is just physical change there outside, but in your head, there's an interaction. A similar thing happens when you [subject] add [interaction] two numbers [object].
    – RodolfoAP
    Apr 5 at 11:12
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I would like to propose something: The Library of Babel.

Imagined by writer Jorge Luis Borges, The Library of Babel is a fictional, labyrinthine catalogue of every possible book and every possible piece of writing that could ever be written. Within it, you could theoretically find the cure to cancer, how you would die, how the universe would end, every one of Shakespeare's works translated to every single language, and your hypothetical mathematical sentence.

Now, this library is purely fictional, but versions of it exist. For example, the Canvas of Babel: an online canvas, with dimensions of 640x416, with each pixel capable of displaying a colour value from 0 to 255. If we assign a random, unimaginably large value to each colour and each space, then theoretically, your hypothetical mathematical sentence exists in the form of an image, located on that website. As in, the image could be decrypted to dictate your mathematical sentence.

So, yes, technically your mathematical sentence does exist, and the information it has exists in the universe. This applies with every other sentence that could ever exist, even if it hasn't been physically written.

We could upscale this further: Instead of a googolplex-long sentence, what about an almost infinitely long sentence? One which exceeds the physical limits of the natural universe? Well, once again, we could theoretically assign random, even larger values to each colour and each space. This would mean that a single image could technically equate to this almost infinitely long sentence. The same could possibly apply for an infinitely long sentence, although that is a much more complex issue.

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  • +1 for Borges allusion.
    – J D
    Apr 5 at 15:01
  • i am confused why you think that an infinte series can be represented in a finite series without loss.
    – andrós
    Apr 6 at 11:04
  • If each colour and location of a pixel represented a limitless series of numbers, then given potential for each pixel to contain an infinite series of numbers, there would be no limit to the length of the numerical sentence it could depict. To describe further: There are 255 possible colours to be used. Each colour pixel could be attributed to 1/255 of an infinite series. With 255 pixels, that means 255/255, aka 100%, of the infinite series can be depicted. Apr 9 at 16:23
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The short answer is it depends on what you mean be "to exist" and "sentence". If existence is taken to be capable of being experienced, then your question is essentially does any arbitrary proposition exist. Philosophers of language tend to use a dichotomy between proposition and utterance, or likewise statement and sentence. Statements and utterances are taken to have physical existence, where as propositions or sentences are taken to have abstract existence. Both of these are metaphysical presuppositions, and not everyone accepts both.

According to linguistic internalism, languages are in some way not objectively real, but are part of the minds of the people who use them and are not external to users of language. If you accept that metaphysical hypothesis, then you have to decide whether existence is validated by a construction or not. So, if you accept that a proposition is a meaningful, abstract object, then you have to decide whether or not a constructivist. Since at least Brouwer, it has become increasingly popular to say that only things that have actually been constructed exist. This epistemological approach is popular with a minority, and might be understood with the difference between potential and actual infinity. (I for one, reject actual infinity.)

Thus the answer you get has no consensus, though mathematicians tend to accept that abstract objects are real and external and discovered, more often than not. In this spirit, all sentences exist and are waiting to be discovered, perhaps by any being capable of language anywhere and at any time in the universe. Some of us embrace much the opposite, saying that sentences only exist in the minds of organisms and their society, and then only when constructed, such as the sentences that are impressed into this physical medium of the Internet. You'll have to decide where along the spectrum you beliefs lie.

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As others have suggested in their answers, it is a matter of semantics- what you are effectively asking is whether it is meaningful to extend the use of the word 'exist' to apply to such things as as-yet un-envisaged sentences. The problem with extending the scope of 'exist' to cover such instances, is that if it applies to one potential but never actualised entity, then it applies to all of them. To take a specific example, it took me three years to write three novels- according to your suggestion, the novels existed all along. If we accept that use of the word existed, then you will have to find another word to describe the other type of existence my books acquired after I had written them, assuming you accept there is an important difference between the two states. Given that, I propose it is simpler to constrain the use of the word 'exist' to its everyday sense, and use the phrase 'potentially exists', or some equivalent word, to refer to the kind of existence you have in mind when you mentioned as yet unimagined sentences.

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1.By definition of sentence, all the sentences can be written down.

2.A Googolplex long sentence still exists on its own, the fact that it might not be containable within this (observable) universe is irrelevant. For example, if all you have is a single sheet of paper do all the numbers bigger than the number of atoms in that sheet of paper do not exist? Or the number of possible arrangements of all the atoms in the universe does not exist just because the power set can NOT be contained within the universe?

Let's make this even simpler, instead of sentence consider that in mathematics it is proven that there are uncountably many numbers that can NOT ever be written down or even computed (i.e. the uncountable cardinality of real line is from transcendental numbers and cardinality of algebraic numbers is still countable).

Yet all those numbers exist, as the real line is continuous. but any infinite discrete universe will not be sufficient to hold digits of all the transcendental numbers.

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At first, I thought too literally about the given example, but I realized OP may have just wanted the sentence to represent some important expression that is absolutely impracticable due to impossible complexity. I believed that a case for distinguishing between virtual and literal existence could be made, but I hadn't found any sense of deeper philosophical meaning, so I looked even more broadly on the example as an apparently insurmountable challenge, a notion simply beyond belief. What occurred to me is that it is beyond our ability to definitively know that the example is truly impossible, and here I found the deeper philosophical meaning I feel is worth pursuing. It may still be beyond my knowledge to site very many relevant works, but questioning the extent of what one can possibly accomplish speaks to a core tenet of Marcus Aurelius.

A googolplex written out in books would require more mass than what we once knew as the observable universe (JWST has expanded our view). If we could record the sentence on Planck-scale media, we could reduce the exponent of 10 by half. We may discover a way to create and use small black holes to compress this media further, or to record it in higher dimensional space. For all we know, we may discover how to process data structures of infinite complexity using quantum computers.

As this is a remote possibility, I will treat it as such. For practical purposes, I'll assume it's not going to happen in my lifetime, because it's not my life goal. If it was my life goal, I would be after it every day, and you would never hear me say it was impossible. Since that is not the case, in most circumstances, I will treat it as practically impossible; if somehow it were up to me to make a decision that would affect the human race, I would treat it much more daringly. My personal belief is to judge these questions more or less strictly according to what the answers are worth, but never to declare anything absolutely impossible, and I would bet you that 9 out of 10 philosophers would agree.

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  • +1 I upvoted on account you make the case for "potential sentences" being given by grammars, but "natural human languages develop almost indifferently to the principles of linguistics" is stated wrong. The principles of linguistics themselves are derived from myriad human natural languages. What they are indifferent to is the prescriptive formal grammars that authorities use to attempt to constrain their development.
    – J D
    Apr 5 at 15:00

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