6

I'm aware of the idea of $:=$ as 'definition, similar to an equality', but I'm confused about how it's used, and whether we mean to define the expression's value for all conditions. What do we mean by definition?

If I define $y:=x^2,$ this makes sense, but what about $$x^2+y^2:=1\;?$$ What am I defining here? How does my 'definition' mean differently from simply stating an equality?

How about $$\sin(x)^2+\cos(x)^2:=1\; ?$$ This always has the value of $1$ for any $x$ so how do I 'define' anything on this, it's either true or false, it's not my definition that makes it true, it is true always.

How do we find truth values for statements given with '$:=$', and what if I make a false equality for all $x$ like $$3x:=x+x$$ how do I 'define' something that can't be true?

ADDENDUM

I'm slightly struggling with these replies, because of my lack of understanding of what 'definition' means.

Writing $x^2+y^2:=1$, which you may technically do, is defining the entire symbol $x^2+y^2$ to be 1. The individual parts of the symbol, $x^2$, + and $y^2$ might have no additional meaning.

and

You are defining something to be something else. If I define a twinkie to be something, any instance of the twinkie is that something. The cream filling or any other knowledge of a similar looking object mean nothing. I could have replaced the $x^2+y^2$ with a smiley face with $x^2$ and $y^2$ substituted for its eyes. It's just a symbol.

What does this mean in terms of 'definition'? Surely we can define a relation (for example the unit circle) without having to somehow verge away from the expression having it's usual mathematical meaning?

Is it ok to make an operation over some definition at the very first time you are defining it?

This question seems to suggest that for some people, using the definition symbol suggests that what is on the left feels like 'some symbol' instead of the mathematical expression it should define.

ryang
  • 44,428
user37577
  • 795
  • 2
  • 14
  • 1
    Comments are not for extended discussion; this conversation has been moved to chat. – Xander Henderson Aug 20 '22 at 19:33
  • 7
    We use the notation $:=$ to differentiate between when we're using the equals sign to assign a definition versus when we use it as $=$ to express an equivalence relation. Basically they $=$ symbol was overloaded with two different meanings so we split them out into two. – CyclotomicField Aug 20 '22 at 20:05
  • This might be helpful. – Torrente Aug 20 '22 at 20:28
  • @CyclotomicField what do we mean by 'definition' in this case? – user37577 Aug 21 '22 at 08:05
  • 1
    You can go your entire math career without ever using the := symbol. Just write full sentences and say things like “Let $y:\mathbb R \to \mathbb R$ be the function defined by $y(x) = x^3$ for all $x \in \mathbb R$.” – littleO Aug 21 '22 at 08:16
  • Of course, At this stage I'm more interested in the idea that using a definition like $x^2:=expression$ somehow means that $x^2$ is just a symbol and not something being squared. – user37577 Aug 21 '22 at 09:04
  • "I'm more interested in the idea that using a definition like $x^2:=expression$ somehow means that $x^2$ is just a symbol and not something being squared" If you're asking if you're allowed to define "$x^2$" however you like: Sure! Calculus has its derivative symbol $\frac{dy}{dx}$, and number theory has its Legendre symbol $\left(\frac{p}{q}\right)$, both of which look like fractions, but aren't. So, feel free to define "$x^2$" as something other than "$x$-squared" ... but be prepared to justify competing with algebraic convention. :) – Blue Aug 21 '22 at 11:56

1 Answers1

5

I'm slightly struggling with these replies, because of my lack of understanding of what 'definition' means.

Then let's go back to definitions (always a good place to start): a definition is a specification of meaning; a mathematical object may be defined by assigning properties to it, by equating it with another object, etc.

and what if I make a false equality like $$3x:=x+x\tag1$$

A literal interpretation of this definition is that it is legitimising replacing every instance of $\text“x+x\text”$ with $3x$ (here, $\text“3x”$ is the subject of the definition).

It could also mean that in the given system, given an input $x,$ the operations $3x$ and $x+x$ return the same result. (There might be an earlier definition giving how the operation $x+x$ works.)

If, additionally, you are working in an arithmetical system, then this definition will lead to contradictions; not because $$3x=x+x\tag2$$ is false equality, but precisely because it is (forced to be) true!

How do we find truth values for statements given with '$:=$'

The equality $(2)$ is—by definition $(1)$— true. When given a definition, within its system, it is true by assumption.

how do I 'define' something that can't be true?

The statement of a definition is taken as true.

Sometimes, a definition leads to contradictions (for example, the statement of the definition turning out to be also false); other times, they are ambiguous, in which case we say that the object being defined is not well-defined.

The point is that creating definitions is a design process, and some designs are shoddy, because issues that crop up are not dealt with, or potential issues are not noticed.

And like all good designs, good definitions commonly undergo multiple revisions: a working definition may have no problem, but I may still keep refining it such that my work flows more naturally, and the final definition might differ drastically from the preliminary one.

$$x^2+y^2:=1\tag3?$$ What am I defining here?

You are defining your points $(x,y)$ of interest precisely as the set of points on the unit circle centred at the origin.

How does my 'definition' mean differently from simply stating an equality?

A definition is a starting point for further work. It may then be invoked in the derivation of a result/theorem or answer.

In the above, I have used your given definition as a starting point in characterising your points of interest as the unit circle centred at the origin.

Please do not conflate a definition and a result/theorem, or a definition and a characterisation.

whether we mean to define the expression's value for all conditions.

It's up to the author to specify the conditions under which their definition applies. In definition $(3),$ I've assumed that you are discussing $\mathbb R^2$ instead of, say, just quadrant $1.$

How about $$\sin^2(x)+\cos^2(x):=1\tag4$$ it's not my definition that makes it true, it is true always.

Equality $(4)$ can be derived from the definitions of the trigonometric functions, so it is mathematically true, so we call it a result/theorem.

In our previous discussion, I alluded to the fact that the trigonometric identity $$\sin^2(x)+\cos^2(x)\equiv1$$ is a theorem/result, not a definition.

An identity in $x$ is simply an equality that holds for all $x;$ in this case, the identity is true by derivation, in another case, it might be true by definition.

Since definition $(4)$ duplicates a known result, it leads to no contradiction, and is merely redundant. On the other hand, $$\sin^2(x)+\cos^2(x):=7\tag5$$ is a bad definition because using it can lead to contradictory results.

how do I 'define' anything on this?

You don't have to define $$\sin^2(x)+\cos^2(x)=1;$$ you merely have to quote (or perhaps derive/prove) it.

ryang
  • 44,428
  • Thanks for this in-depth answer, do you understand the idea of interpreting $x^2=(expression)$ with $x^2$ as a single symbol and not something squared? – user37577 Aug 21 '22 at 10:05
  • Yes, refer to my second paragraph, which starts with, "A literal interpretation". And you may have missed the implicit point in my third paragraph telling that the concept of squaring is arithmetical: when you read $x^2$ as "something squared", you are already suggesting that the system is arithmetical. – ryang Aug 21 '22 at 10:28
  • Thanks for the responses – user37577 Aug 21 '22 at 11:12
  • You're welcome, and, by the way, this previous answer that I wrote may be of interest. – ryang Aug 23 '22 at 05:15
  • BTW, please don't keep frivolously closing accounts then creating new ones; there is a clear and educational continuity in the sequence of your numerous posts, from those from that 1st account, through the 2nd account, etc., to the current one. By 'educational', I mean for yourself (in fact, I feel that you don't refer enough to previous Answers given to you; this impedes the evolution of your understanding) and for other users of this site. Even when your understanding has evolved, especially since your interest is the intricacies of technical issues, each revisit still reveals new layers. – ryang Aug 23 '22 at 05:30
  • I kept closing it, then realizing I had more questions, the technical intricacies are the things that I ask most because they are the things that aren't taught much/well in formal education, unfortunately they can be difficult to express! – user37577 Aug 23 '22 at 07:35