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Equality sign - Different meanings in mathematics

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Since my first year on school, until the last year of high-school I was always taught one type of equality sign: = , to be used in all mathematical statements. On the first year in my university I was amazed to discover that some more intelligent equality signs exit, but are rarely used. The different signs are denoted by an added symbol on top of the "=", and they express the different meaning of the mathematical statement.

When I learned about them I realized just how many problems and misunderstandings could have been solved if kids would be taught mathematics using the different signs.

These are the types I know of:

  • "?" on top of an "=" : Checking to see if the relation described by the equation is true (if eventually it gives 1=1)
  • "!" on top of the "=" : Demanding that a certain mathematical relation exist, and discovering what are the conditions that have to be met for that relation to be true
  • "/\" (a little triangle) on top of the "=": Equality that comes from definition, which is taking a certain mathematical entity (such as "x^2+8x+9") and naming it (like "f" or "g").

Generally, the meaning that "=" gives to both sides of an equation is "Those are the same", or "This statement is true".

The problem created when teachers only use "=" is that all other types of mathematical statements are understood in the exact same way, to be true statements, which is not the exact meaning of the statement.

So when I want to find the value of extreme points of a function, by demanding that the derivative equals zero, I am writing it as if I mean to say that the derivative equals zero (as in: "this is true"). This creates the question in the young mind of the student: "But how do you know it's true?" Because they interpret it in the same way like "2+2=4".

Same thing goes for definitions: While the question "How do you know this is true?" applied to "7+1=8" or "7*6/3=14" has a meaning, it is meaningless to try to apply it to "f=x^2+8x+9", since "f" is just a name, and not a result.

[if anyone understands the statement I was making in my last sentence, please let me know about it, because I've found that some people have a hard time understanding what I mean by that, and it would be great to see that more people are capable of understanding that statement.]

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  • "/\" (a little triangle) on top of the "=": Equality that comes from definition, which is taking a certain mathematical entity (such as "x^2+8x+9") and naming it (like "f" or "g").

There is another variation of this: 1 additional horizontal bar above the equal sign.

Same thing goes for definitions: While the question "How do you know this is true?" applied to "7+1=8" or "7*6/3=14" has a meaning, it is meaningless to try to apply it to "f=x^2+8x+9", since "f" is just a name, and not a result.
It does have a meaning and an answer: "Function f is like that because it was given as such."

Context matters here. Compare this to a form, where "f=x^2+Ax+9", and we wish to discover the value of 'A', then the question becomes valid yet again: is f=x^2+8x+9 or f=x^2+9x+9, etc. ?

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It does have a meaning and an answer: "Function f is like that because it was given as such."

The question is inapplicable and has no meaning when asked about an equality of definition, but it does have an answer.

Suppose you ask "Is it true that "7+1=8"? "

The answer is: It is true because 1 represents a single unit, and since the unites cannot destroy one another or reproduce when put together, they are additive, in a way that 1+1=2, 1+1+1=2+1=3, n+1=(n+1) etc'.

Suppose you ask "Is it true that "f=x^2+8x+9"?" The answer is: "Well, it depends on the definition. Is "f" defined to be "x^2+8x+9"?"

So now I answer "But, what do you mean, "it depends on the definition"?, this IS the definition."

"Hmm... I see. So tell me, is it true that "f=x^2+8x+9"?" (The other nagging person asks again)

"As I said, it will only be true if it was DEFINED that way. There is nothing metaphysical that causes "f" to be "x^2+8x+9". And that is why you cannot ask whether a definition is true or false. It is, a Definition: it is what allows you to ask questions of true or false later on".

"Hmm :pirate: . I see. Thank you, it seems so clear now I wonder how come I didn't think about it myself."

Context matters here. Compare this to a form, where "f=x^2+Ax+9", and we wish to discover the value of 'A', then the question becomes valid yet again: is f=x^2+8x+9 or f=x^2+9x+9, etc. ?

You cannot discover the value of 'A' from that: you don't have enough restrictions. You have 2 degrees of freedom. What you did was actually to define a different "f": a function of two variables.

IF you wrote down f=0 or x^2+Ax+9=0 then you would have 1 DOF, and if you would write f=0 & x=1 you would be able to get values for all the parameters (x,A and f). But without giving more equations I don't see how you can try to "solve" "f=x^2+Ax+9".

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But without giving more equations I don't see how you can try to "solve" "f=x^2+Ax+9".
I meant that this was given within a context of a problem where more information is given, enough to at least say something about 'A'. And in such context, f would be describing something, and the above question of its validity would become valid.
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I meant that this was given within a context of a problem where more information is given, enough to at least say something about 'A'. And in such context, f would be describing something, and the above question of its validity would become valid.

Of course that "f" would be describing something. I don't understand your point.

What "question of validity" are you talking about?

Do you understand what I said in my last post, about the difference between "1+1=2" is true and

"f/\=x^2+8x+9" is a Definition (and that you cannot ask if it is true until AFTER the definition has been made)?

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Another thing I want to add is the difference between identification and definition.

suppose I say "a horse is a chicken with no wings".

I actually said: horse = chicken with no wings

Since horse relates to a certain animal which is already defined, and a chicken with no wings relates to another animal (in a certain physical state) This would be a case of wrong identification.

But if I say: "definition: horse - chicken with no wings"

I actually said: horse /\=chicken with no wings.

In this case I used an already existing name ("horse") as a name for a new concept. This is a very bad choice of a name for a concept because the name is already used for something entirely different, but it is possible to use one name for different concepts. This would make my definition pretty crappy to use, and very uncomfortable but the question of whether or not it is true is inapplicable here. Something can be true or false only if it an attempt at an identification of reality. If I visualize a horse (creature that looks like this) and I see a chicken with no wings and say "those are the same" then I misidentified: my statement contradicts reality.

But when I define something I am naming it. And as a name it cannot be true or false. "f=x^2+9x+8" is not a statement of truth (or false) unless "f" was defined to mean that, or defined to mean something else, first.

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  • 2 weeks later...
Of course that "f" would be describing something. I don't understand your point.
Let's say we are looking to describe something. Something we do not know completely yet. Let's say we use name 'f' to describe a property of that something. Let's say we have found through analysis that 'f' must be of the quadratic form. Now, this gives only a portion of the answer.

Read on to the next quote for continuation.

What "question of validity" are you talking about?
And the question of validity is now to find the proper coefficients for 'f' that match the properties of something that we are analysing. "f=x^2+Ax+9" means we found some of them, but one is still missing, 'A'. Once we find 'A' we can check if our caclulations were correct, i.e. 'f' does indeed properly describe the property.

This entire process makes 'f' either true or false. This depends if it matches reality or not.

This is what I meant with

Context matters here. Compare this to a form, where "f=x^2+Ax+9", and we wish to discover the value of 'A', then the question becomes valid yet again: is f=x^2+8x+9 or f=x^2+9x+9, etc. ?

Do you understand what I said in my last post, about the difference between "1+1=2" is true and

"f/\=x^2+8x+9" is a Definition

Is this difference that one can not ask if a definition is true or not?

... and that you cannot ask if it is true until AFTER the definition has been made?
I don't see the point here. Does it mean that one cannot ask if something is true or false before the judged statement is spoken first? If so, I don't see the point of it. Of course, one cannot analyze something before it is given for the analysis. Did this statement come as a response to something in my post?
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I want to explain first the background for this current discussion:

According to Objectivism, definitions are statements of truth. This is because, according to Objectivism, definitions are statements of identification of a special kind: identifications of essentials of a concept (with genus and differentia).

On the other hand, I too have formed my own definition of definitions, which is different from the definition that Objectivism has for definitions. I find some problems with the Objectivism definition of definition as I will explain later.

According to my view, definition is a statement that attaches a symbol to a concept by providing a precise, full description of a concept (*).

According to how I view definitions, a definition has 2 parts: 1 is the symbol, and the second is the description. When I said that definitions are the things that allow us to communicate I was relying on my definition of definition: If the meaning of symbols is not shared by people, they cannot communicate. They would interpret the symbol "man" in different ways.

According to my view, a definition has tremendous significance, because it is a corner-stone that is used to form ideas in the form of verbal statements, and therefor a symbol must have a very precise meaning attached to it for the idea to be communicable.

(*) A precise description of a concept may be achieved by pointing out to some mental integration that is known to be shared among humans (for example, the general attributes of a cat is something that is probably perceived in the same way by different people).

In science when the things you discuss become less tangible (in the sense of being able to use our senses to directly perceive the concretes) the significance of providing a precise verbal description increases, since the concept cannot be describe by referring to some shared mental integration that can be achieved by directly perceiving it's concretes in the physical world.

For convenience, I would define (according to definitionI) the two meanings of definitions, that I will use:

DefinitionO would be the definition of Objectivism (and if you have anything to add to how I described it above, please do so).

DefinitionI would be the meaning of definition that I have been using, as I described it above.

I hope you don't mind me using these words, because this is the best way I can express myself.

Notice also that according to definitionI, "definitionI" is a symbol, while according to definitionO, "definitionI" has to be a concept. You may ask, "but how do you know that the symbol "definitionI" represents the concept "definitionI"? The answer that definitionI gives is automatic, but the answer definitionO gives is: "I don't know, it was done is a process that is not a part of the process of defining something".

I want to take this step by step to avoid misunderstandings, so for now I am just going to answer your post.

Let's say we are looking to describe something. Something we do not know completely yet. Let's say we use name 'f' to describe a property of that something.

Hold it right there. What would you call this process, of using the name 'f' to describe the property of that something? (me thinking...)

  1. Do you consider this to be an ostensive definition?
  2. Do you consider Ostensive definitions to be statements of truth, or just formal definitions to be statements of truth?
  3. How is it true to use the name 'f' to describe that something? (in other words, is "definitionI" a statement of truth?)

Edit: a clarification about question3: The way I see is, an ostensive definition and definitionI are of the same type (name giving + description). So in fact question3 only kicks in if you give a positive answer to question2.

Edited by ifatart
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Which specific meaning is to be attributed to any particular use of the equality sign is, usually, perfectly clear from context. For example, an initial statement

f = x^2 + 8x + 9

can only be a definition for f, not an equality test or equality assertion. Any subsequent use having a similar appearance can only be an equality assertion, not a definition. The relevant contexts in this case are the answers to the question has f previously been defined?

The evaluation of the points on the curve f(x) having derivative zero is

C(f) = { (x, f(x)) | df/dx(x) = 0 }

The outer equality is an equality assertion (where implicitly I have defined C to map a function to its zero-derivative points, and f is defined as before). The inner equality is an equality test, meaning, it yields true or false; it is neither an assertion nor a definition, as its context informs us.

Often, this question is put into words as follows: find the values of x for which

df/dx(x) = 0

and only this equality test is given explicitly in mathematical notation. Again, what this equality symbol means is clear from the context.

It is only in the case of equality assertions that the question how do I know this? can be asked. When to ask this question, and when not to, is generally clear from the context. As to your last comment, if f had previously been defined, then one could ask the question; but if not, we assume that the equality sign signals a new definition of f, and the question cannot then be asked.

Edited by y_feldblum
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According to my view, definition is a statement that attaches a symbol to a concept by providing a precise, full description of a concept.
Then how is it different from 'word' ? And doesn't that imply that different languages have different definitions of the same thing?
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  • 2 weeks later...
Which specific meaning is to be attributed to any particular use of the equality sign is, usually, perfectly clear from context.

It can be clear from the context, in the same way that eliminating vowels from certain words can still leave the word recognizable.

When math is taught for the first time, it helps a lot to be introduced to these symbols. I myself use them because it makes things easier for me, because I like the meaning of a sentence to be as accurate as possible, and when I read some exercise I wrote a few month ago, it makes the understanding of what I've done quicker. I guess it is personal preferences.

The evaluation of the points on the curve f(x) having derivative zero is

C(f) = { (x, f(x)) | df/dx(x) = 0 }

The outer equality is an equality assertion (where implicitly I have defined C to map a function to its zero-derivative points, and f is defined as before). The inner equality is an equality test, meaning, it yields true or false

It doesn't yield true or false: it yields values of x for which it is true. I agree with the second part of your sentence: "it is neither an assertion nor a definition, as its context informs us.

Often, this question is put into words as follows: find the values of x for which

df/dx(x) = 0"

It is only in the case of equality assertions that the question how do I know this? can be asked. When to ask this question

...

if f had previously been defined, then one could ask the question; but if not, we assume that the equality sign signals a new definition of f, and the question cannot then be asked.

All fine and well in mathematics.

What about language? In Objectivism definitions are statements of truth (Because they are identifications of reality, according to Objectivism). Obviously, this definition of 'f' in this case is not a statement of truth.

I was told that 'f' is not a formal definition, so just because it's definition is not a statement of truth, does not say anything about formal definitions. But I've been thinking... Why isn't this definition of 'f' a formal definition? the genus being a mathematical function, and the differentia being a second order polynomial with coefficients 1, 8 and 9.

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It can be clear from the context, in the same way that eliminating vowels from certain words can still leave the word recognizable.

Which particular meaning to ascribe to an equality sign is clear from the context, in the same way that it is clear from the context which particular meaning should be ascribed to a word having more than one dictionary meaning.

When math is taught for the first time, it helps a lot to be introduced to these symbols. I myself use them because it makes things easier for me, because I like the meaning of a sentence to be as accurate as possible, and when I read some exercise I wrote a few month ago, it makes the understanding of what I've done quicker. I guess it is personal preferences.

Very often, one finds statements such as "Let f(x) = ..." - the let is a clear and explicit contextual clue that the equality sign means definition or assignment. (Oftentimes the word define replaces the word let.) While it's not necessary, since the context is clear, using such words can be clearer than using sutble variants of the equality sign.

It doesn't yield true or false: it yields values of x for which it is true. I agree with the second part of your sentence: "it is neither an assertion nor a definition, as its context informs us.

Look again. The entirety of the statement between the braces yields some values of x, but the equality test yields either true or false for every value of x.

What about language? In Objectivism definitions are statements of truth

That's not correct. At any rate, there's a distinction to be made between an assignment-definition in mathematics and the definition of a concept in philosophy.

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