Why is it hard to define something?

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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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  • #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.
 
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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.
I don't think this is because of languages being circular. Can you please explain?
 
  • #4
AlephZero
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I don't think this is because of languages being circular. Can you please explain?
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....
 
  • #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.
 
  • #6
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Arildno said:
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.
I don't follow this. What processes do you mean?
 
  • #7
arildno
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I don't follow this. What processes do you mean?
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.
 
  • #8
Office_Shredder
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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.
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).
 
  • #9
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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.
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.
 
  • #10
Dembadon
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I don't think this is because of languages being circular. Can you please explain?
From the following wiki: Circular definition

The 2007 Merriam-Webster dictionary defines a "hill" and a "mountain" this way:

Hill - "1: a usually rounded natural elevation of land lower than a mountain"
Mountain - "1a: a landmass that projects conspicuously above its surroundings and is higher than a hill"

Merriam-Webster's online dictionary provides another example of a circular definition with the words "condescending" and "patronizing:"

Main Entry: condescending
Function: adjective

1 : showing or characterized by condescension : patronizing

This definition alone is close to suffering from circular definition, but following the definition train:

Main Entry: condescension
Function: noun

1 : voluntary descent from one's rank or dignity in relations with an inferior
2 : patronizing attitude or behavior

Looking up the word "patronizing" then gives us:

Main Entry: patronize
Function: transitive verb

1 : to act as patron of : provide aid or support for
2 : to adopt an air of condescension toward : treat haughtily or coolly

In short: the two words define each other.
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.
 
  • #11
sometimes circular
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.
 
  • #12
Dembadon
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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.
I don't understand the "human" distinction you've given language. Aren't all languages human constructions?

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.
This is impossible. For something to have a definition, it had to have been encountered, otherwise it would not have been assigned a definition.
 
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  • #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.
 
  • #14
Dembadon
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^ By "human" I mean natural, as opposed to "formal" languages, like the language of all palindromes over a binary alphabet.
Ah, I understand now.

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.
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".
 
  • #15
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Yep, "something" is a hard word to define. What does it mean to be something?
 
  • #16
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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?
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.
 
  • #17
Dembadon
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Yep, "something" is a hard word to define. What does it mean to be something?
Something is the complement of nothing. :biggrin:
 
  • #18
Chronos
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The problem with definitions is they require axioms [assumptions] - like 1 = 1.
 
  • #19
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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.
Galileo defined math quite beautifully- "Mathematics is the language with which God has written the universe."
Edit: Or was it Pythagoras?
 
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  • #20
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The problem with definitions is they require axioms [assumptions] - like 1 = 1.
Yes, language requires axioms but still there exists a definition for every word. Why?
 
  • #21
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That's the purpose of language isn't it. If the words were meaningless language would be redundant.
Definitions help giving a meaning to a word for example if I were to say Floccinaucinihilipilification, it wouldn't mean much to most people- Language defines itself by definitions- the word is defined as the act of estimating as worthless.
 
  • #22
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That's the purpose of language isn't it. If the words were meaningless language would be redundant.
Definitions help giving a meaning to a word for example if I were to say Floccinaucinihilipilification, it wouldn't mean much to most people- Language defines itself by definitions- the word is defined as the act of estimating as worthless.
I meant that even though it requires for a language to have certain axioms, still there exist a definition for every word. So I was asking, how is that possible?
 
  • #23
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Take an axiom, define it- define its definition- rinse and repeat enough times you get back to the same axiom or another one.
Think of it as a web with nodes as words and then you will find to explain some word (axiom) you will get back to eventually another or same axiom.
 
  • #24
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Take an axiom, define it- define its definition- rinse and repeat enough times you get back to the same axiom or another one.
Think of it as a web with nodes as words and then you will find to explain some word (axiom) you will get back to eventually another or same axiom.
I would love a graph showing how all our definitions are circular (I get the point of how languages are circular that someone made earlier). Can I find such a graph on the internet or somewhere else?
 
  • #25
harborsparrow
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Even the concepts that words are pointers to are sometimes so complex that they can defy easy definition. For example, what is a room? In order for there to be a room, there already have to exist in your lexicon a whole lot of other words and concepts. Even then, it's difficult to be precise--when does a box-like space stop being a room and become a box, for example?

Philosophers expend a lot of energy worrying about this stuff. I don't.
 

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