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Why is it hard to define something?

  1. Aug 29, 2013 #1
    Why is it so tough to give a definition for something even if we know what it is?
    For e.g. Math:- There is no proper definition of math, many people argue about what exactly is it. But still, even if we do not have a proper definition, we know what math is and what it's not. When calculus was invented, everyone knew it was math.
    So we can tell what is what but can't give a proper definition. Why is that?
     
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  3. Aug 29, 2013 #2

    Dembadon

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    Languages (i.e. English) are imprecise and sometimes circular. They work well enough for everyday communication, but I don't believe they'll ever be as precise as mathematics.
     
  4. Aug 29, 2013 #3
    I don't think this is because of languages being circular. Can you please explain?
     
  5. Aug 29, 2013 #4

    AlephZero

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    When I was at university, one of my math tutors gave a good example. He had decided to start teaching his 6-year-old kid some of the basic ideas of mathematics. One day they were walking along the river bank, and some of the college rowers were practising. He pointed out to his kid the fact that there were 8 rowers in the boat, and also 8 oars, one for each rower, and that this was called a "one to one correspondaece."

    A few days later they were on the river bank again, and he asked his kid if he could remember what a one-to-one correspondence was. The answer: "Yes dad, it's a sort of boat".

    It's easy to define stuff be reference to what you already know, but the hard part is getting started when you don't know anything much....
     
  6. Aug 29, 2013 #5

    arildno

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    Because for most of the existence of the human race, definitions have been largely unnecessary.
    Can you define, and give a precise verbal description of the people you love, perhaps?
    Or of love as such?

    Our capacity for precision in defining something is a rather faulty by-product of the processes of evolution that by themselves gives no adaptive advantages.
     
  7. Aug 29, 2013 #6
    I don't follow this. What processes do you mean?
     
  8. Aug 29, 2013 #7

    arildno

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    How should I know?
    What I do know is that the ability for precisely verbally defining phenomena, events or whatsoever has had absolutely zero adaptive advantage. In contrast to non-verbal learning skills.

    Thus, it seems likely that abstract thought is more of an interesting by-product of other evolutionary processes, than something whose development was directly refined by evolution.
     
  9. Aug 29, 2013 #8

    Office_Shredder

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    If everyone agrees with what is and is not math then they cannot argue about the definition (the only way they can argue is if one person's definition says X is math and anothers says X is not).
     
  10. Aug 29, 2013 #9
    If you don't consider the ability to communicate verbally an advantage, I don't suppose you'd mind waking up completely deaf tomorrow. After all: you'd still have your non-verbal learning skills.

    If Og, the caveman, couldn't run back to his tribe and communicate to them that he'd just spotted a herd of bison, they would not eat well and not flourish. The gestures or words he used had to have a specific definition accepted by all.
     
  11. Aug 29, 2013 #10

    Dembadon

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    From the following wiki: Circular definition

    It makes it very difficult, and almost impossible in some cases, to obtain precision when languages contain words that rely too heavily on either themselves or other words which might have ambiguous, subjective, or circular definitions.

    It's also worth noting that I believe what I've mentioned is a contributor, not the absolute cause, to the issue you've presented.
     
  12. Aug 30, 2013 #11
    Either there are some words which are left undefined, or the language is circular. Assume that all words are defined and the language isn't circular. Consider any word in the language. By assumption, the definition of this word consists of some set of words which we haven't encountered yet. The definition of each of these words must also consist of some words which he haven't encountered yet, etc., ad infinitum. This is a contradiction since languages (human languages, like English) consist of finitely many words. Therefore, (finite human) languages must either contain words without definitions, or be circular.

    Conjecture: (Finite human) Languages are circular and contains words without (good) definitions.
     
  13. Aug 30, 2013 #12

    Dembadon

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    I don't understand the "human" distinction you've given language. Aren't all languages human constructions?

    This is impossible. For something to have a definition, it had to have been encountered, otherwise it would not have been assigned a definition.
     
    Last edited: Aug 30, 2013
  14. Aug 30, 2013 #13
    ^ By "human" I mean natural, as opposed to "formal" languages, like the language of all palindromes over a binary alphabet.

    Regarding the second point, I suppose "encountered" is misleading. I suppose a better way to say it would be "listed so far during this exercise".

    So, for instance, if we start with "tree", we might use the definition "a plant with a woody trunk and leaves". This definition is valid since it doesn't reference "tree". Next, we might consider a definition of "plant", and so on. Eventually, since natural/human languages like English are finite, we must eventually encounter the same word twice, or find a word that has no definition.
     
  15. Aug 30, 2013 #14

    Dembadon

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    Ah, I understand now.

    I see what you're saying now. At this point, I believe we need to define what we mean by precise (oh, the irony :biggrin:). If we are going for an infinite level of precision, then your method is probably the most rigorous. But I'm assuming the OP has some limit to the level of precision he/she desires for a sufficient definition. Therefore, assuming a finite level of precision, we should be able to obtain a satisfactory definition without having to traverse all possible paths/relations in a word's "definition graph".
     
  16. Aug 30, 2013 #15
    Yep, "something" is a hard word to define. What does it mean to be something?
     
  17. Aug 30, 2013 #16
    Not to be overly pedantic... but first don't we have to define what you mean by "definition"? Because description and definition are similar but not the same.

    You can define a word from within that language itself, you can define it by associating it to a word in another language, you can define a word by its a association with a physical object, or you can get associate with some philosophical ideal... in the latter cases, language is used to describe not define.
     
  18. Aug 30, 2013 #17

    Dembadon

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    Something is the complement of nothing. :biggrin:
     
  19. Aug 31, 2013 #18

    Chronos

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    The problem with definitions is they require axioms [assumptions] - like 1 = 1.
     
  20. Sep 1, 2013 #19
    Galileo defined math quite beautifully- "Mathematics is the language with which God has written the universe."
    Edit: Or was it Pythagoras?
     
    Last edited: Sep 1, 2013
  21. Sep 1, 2013 #20
    Yes, language requires axioms but still there exists a definition for every word. Why?
     
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