(late edit: I have come to the conclusion that no words (or claims) have 'direct' reference to reality since all words are about people's conception. Words reference particular concepts in people's minds (i.e. the people who used or are using the words). People's conception can then directly (from memories of empirical experiences) or indirectly (though other conceptions) reference reality. By providing definitions of words we try to understand the same concept but in the light of words referencing concepts which are more familiar to us. Preferably concepts we have real life experience with i.e. those that have a more direct references to personal experiences for us. We often continue defining the definitions until our conceptions reference direct empirical experience i.e. some evidence that what we are trying to conceive have some effect in reality according to our past experiences. This is why we can never fully understand things until we have real life experience with it.)

I just answered this question about 'Whether a definition is a proposition?' by arguing that it is. My argument is based on one of my most fundamental ideas. It is an argument inspired by Quine, which leads to the claim that there is no fundamental difference between a fact and a definition.

"Like other Analytic philosophers before him, Quine accepted the definition of "analytic" as "true in virtue of meaning alone". Unlike them, however, he concluded that ultimately the definition was circular. In other words, Quine accepted that analytic statements are those that are true by definition, then argued that the notion of truth by definition was unsatisfactory. " (http://en.wikipedia.org/wiki/Willard_Van_Orman_Quine#Rejection_of_the_analytic.E2.80.93synthetic_distinction)

I will put my argument underneath so you know what I'm talking about, but my question is basically whether it is true or not that both definition claims and fact claims can (and therefore should) be verified and falsified against reality; does anyone know about any useful definitions (useful words) which have no direct or indirect reference to reality (i.e. the physical world); what words, that are not directly or indirectly verifiable against reality, not even hypothetically, do we find useful?

"... The way I understand the theory, or my spin on it, is that ultimately all definitions have to be verified and falsified in the physical world just like facts do, since they are ultimately about the same thing; they are about our experiences of reality. When you define words with other words you are at the same time making claims about what is true (i.e. propositions). (side note: a made up a story can have definitions which are not about reality, but they are still about our experiences of reality since they are about a story which exist in a book, online or memory etc.) When you claim that e=mc2 you are both saying that mc2 is a useful definition of energy and that it is true that the function of energy can be determined by these measurements. A more straight forward example; when you say that the earth is round you are at the same time saying that round is a good definition of the shape of the earth and that it is true that the earth is round (it is actually no way near round since we have high mountains and deep valleys). When you attack the argument it can be about definition or fact, but if you change one you will have to change the other in order to stay coherent. If you say that 'almost round' is a better definition of the shape of the earth, then it is also true to you that the earth is 'almost round' and not round anymore. If you conclude that it is true that the earth is round then you are also saying that the shape of the earth is a useful definition of round.

An interesting consequence of this is that 1+1 is not necessarily 2. In order to claim that 1+1 is a useful definition of 2 you have to prove it in the physical world. We have a lot of proof since we have added up a lot of apples and bananas throughout history, but there are cases when it is not true. If you add 1 pile of sand to 1 pile of sand you don't get 2 piles of sand, you only get 1. So 1x+1x is not always 2x and so it is not always true that 1+1 is 2 and so 2 is not always a useful definition of 1+1 in the same way that the shape of the earth is not always a useful definition of round (even though sometimes it is).

Summary: all definition are propositions because unless you can verify or falsify something which they are supposed to directly or indirectly be about they are not useful definitions because they ONLY try to describe something which does not exist i.e. something which does not have the slightest present or historical reference to reality. What use would 1+1=2 be if it had never been used to verify or falsify any real world events?"

  • I can see you copied your answer on another question in here. What I don't see is what your question is. I suppose that's why the answer below isn't much of an answer. – iphigenie Mar 20 '13 at 14:32
  • "whether it is true or not that both definitions and facts can (and therefore should) be verified and falsified against reality; does anyone know about any useful definitions (useful words) which have no direct or indirect reference to reality (i.e. the physical world)" – Kriss Mar 20 '13 at 14:44
  • If all words directly (e.g. dog, fast, tall) or indirectly (e.g. fear, lying, attraction) reference reality then nothing can be true by definition alone. Every claim can (at least in theory) be verified and falsified at a particular time in space. It can be true or false what you define as fast and lying. – Kriss Mar 20 '13 at 14:51
  • The meaning of the word bachelor or married might change in the future so that is no longer true that 'all bachelors are unmarried'. – Kriss Mar 20 '13 at 14:59
  • @Kriss: Have you come across temporal logic? – Mozibur Ullah Mar 21 '13 at 1:20

This answer expands on your example that 1+1 may not equal 2. Traditionally of course it does, but there are more modern interpretations where what you say is correct and this is rooted in Russels type theory or in more contemporary idiom category theory.

Whats missing is the notion of type/context/category. To interpret 1+1 we must say what the abstract symbol 1 and + refer to. This information is given by type. (Not every context will support addition - but in our two examples there is such a notion). When the symbol 1 refers to the usual '1', and + is the usual addition in the type or category of integers then 1+1=2; we can write this in 1:int+1:int=2:int

But this doesn't have to hold. In the trivial category (where there is just one object, usually identified as 1), then 1+1 must equal 1 as there is that single object 1.

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Mathematical definitions are useful and if at all only refer to 'mathematical reality' (in the sense of Platonism), not to the physical world.

You seem to suggest about mathematics what Frege called "Pfefferkuchenarithmetik", it has been suggested before but didn't win a lot of followers. I can infer that 1+1=2 from a set of axioms I have set up, e.g. the Peano axioms. Whether that's useful or not is a highly subjective notion. I believe it is. Take the following example. I ask you for 1 dollar, and later I ask you for another dollar. Wouldn't it be useful to know that in total you'd loose two dollars, even if you decided to never give me anything?

Do not forget that there are no completely perfect mathematical objects in physical reality - no points without extension, no perfectly straight lines, no triangles made of perfectly straight lines, and so on. Yet we need mathematics not just for understanding the world in the first place, but for all kind of intelligent planning where not all considered alternatives are realized.

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  • I read a bit about Frege and are very confused :) "The drawers being green is an objective fact, grounded in the external world. But this is not the case with 5. Frege argues that each drawer is on its own green, but not every drawer is 5. ... Frege ... takes numbers to be objects" (en.wikipedia.org/wiki/The_Foundations_of_Arithmetic) are you saying that this is what I'm doing? Why couldn't the Peano axioms be verified in reality? If you change all variables for real objects the claims will turn out true, wont they? How many dollars I gave is easily verifid. – Kriss Mar 20 '13 at 14:38
  • I'm not sure what you are saying over all. But I agree with you that mathematics can be done without constantly referring back to reality, but for it to aid in intelligent planning it will in the end have to correspond with some real situations or event. All plans don't have to be realized, but as long as there are reasons to believe they could be they have reference to reality which can be tested. – Kriss Mar 20 '13 at 14:42
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    I don't know what you mean by "reference to reality", and for me reality encompasses more than what is empirically Given anyway. In any case, I do not see any reason why a justification of "1+1=2" in the form of a proof on the the basis of Peano axioms would need to rely on any empirical test. Just because we use empirical concepts while justifying a claim does not mean that the justification itself is a posteriori. – Eric '3ToedSloth' Mar 20 '13 at 16:01
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    I think I mean that we always use both rational and empirical method to prove a claim. Because if 1+1=2 had never been verified (if we never saw something we called 1 thing being added up with another thing and then we saw that we now had 2 things) I doubt 1+1=2 would be meaningful to anyone. – Kriss Mar 20 '13 at 16:12
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    There are infinitely many integers and in my opinion any addition of two them is perfectly meaningful, although you can never verify all of these additions if by verifying you mean giving some empirical example or even just doing the calculation. To return to your question about usefulness, in a sense you have some company in the people who considered imaginary numbers totally useless - until they turned out to have crucial applications in physics centuries later... We must stop here, though, as this is not a discussion forum. – Eric '3ToedSloth' Mar 20 '13 at 17:02

The supporting evidence using '1+1=2', specifically:

What use would 1+1=2 be if it had never been used to verify or falsify any real world events?"

is simply not true. This is the purpose of formal proof employing layers of logic all of which are entirely useless sans a single, undeniable, provable truth at the very empirical base of the logic.

The fact definitions must be adequately described, assumptions and givens must be clearly presented, and use of symbols must be clearly laid out is beside the point entirely as once these tasks are complete, they need not again be referenced i.e., they are given.

One plus one (1+1) is ALWAYS two (2) no matter what universe or meta-reality you may call home.

Of course one may argue the very inception of such logic would not have occurred without direct observation of the observable universe. This argument, however, does not change the fact 1+1=2 is always true.

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