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Tao (Analysis I, 2022, 4e, p. 285):

(Symmetry axiom). Given any two objects $x$ and $y$ of the same type, if $x = y$, then $y = x$.

What does Tao mean by "same type"?

If $x$ and $y$ are not "of the same type", how can we even have $x=y$?

Is there an example of two objects $x$ and $y$ that are not of the same type but $x=y$? If not, why does Tao even bother stating that they are "of the same type"?

Prem
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  • "Same type" means, e.g., Reals, or Complexes, or Matrices, or Spinors, or Sheafs, or... – David G. Stork Jun 30 '23 at 03:36
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    Tao explains this on the previous page: Given two such objects x and y, the statement x = y may or may not be true; it depends on the value of x and y and also on how equality is defined for the class of objects under consideration. For instance, as real numbers, the two numbers 0.9999 . . . and 1 are equal. In modular arithmetic with modulus 10 (in which numbers are considered equal to their remainders modulo 10), the numbers 12 and 2 are considered equal, 12 = 2, even though this is not the case in ordinary arithmetic." Also see the last paragraph on pp. 285 – Andrew Jun 30 '23 at 03:43
  • @AndrewZhang: But in that example, aren't $12$ and $2$ of the same type? –  Jun 30 '23 at 03:48
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    12 is a symbol. It has a particular meaning if you identify 12 as a real number, and a different meaning if you identify 12 as an element of $\mathbb Z_{10}$. The point Tao makes is that equality, strictly speaking, only makes sense if both objects are of the same type. Indeed, see the last paragraph on pp. 285, I think the explanation is very good. – Andrew Jun 30 '23 at 04:30
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    It is not clarifying OP Doubt : When $x$ & $y$ are not "Same type" & $x=y$ is not true , then the Implication will hold , @AndrewZhang , Why then is it necessary to state "Same type" ? When we already agree that Equality is valid for "Same type" , then the Axiom need not redundantly state that. – Prem Jun 30 '23 at 05:27
  • @Prem I think the last paragraph on pp. 285 addresses this. – Andrew Jun 30 '23 at 06:27
  • Possibly related – Accelerator Jun 30 '23 at 06:38
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    In Previous Page "A.7 : Equality" , Tao says "Equality links two Object of Same type" , then later Tao says "x=y is automatically false when we have Different types" : Over-all , the Axiom seems to be redundantly stating what is agreed , @AndrewZhang , which might be over-sight. – Prem Jun 30 '23 at 07:51
  • Some unsolicited advice: while I consider Analysis I to be the greatest technical book ever written, I found the appendix on logic very confusing despite having some experience with it. If you have no experience with logic, I recommend learning the basics of propositional logic and truth tables from another source. The appendix is still worth reading, as it contains interesting insights into types, equality , vacuous statements, etc. But the whole section on implication, negation, etc. could have been greatly simplified with truth tables – Dr. Momo Jun 30 '23 at 11:01
  • I think it is a matter of language instead of logic, as when we say "a function f is increasing if..." or "the determinant of a matrix A is...". Of course, we don't need the expressions "a function" (because it has to be a function in order to be increasing) and "a matrix" (because it has to be a matrix in order to have a determinant), but it seems some sentences sounds better with logically unnecessary words. – Pedro Jun 30 '23 at 11:19
  • That may be true in general & in normal languages , not in Precise Mathematics & logic , where redundancy is shunned , @Pedro , Eg A Number is Positive is very Different from A Matrix is Positive. Variables can Increase , not just functions. A Square Matrix is Different from A Matrix & from A Square. Norm , Matrix Norm , Vector Norm are all Different. In Precise Math & logic , every word counts & unnecessary words are avoided. Discussions may have extra words , not Axioms. – Prem Jun 30 '23 at 11:54
  • I missed out 2 Points earlier , about your Comment "matter of language instead of logic" , @Pedro , Tao is Comparing & Contrasting Same type & Different type , in Elaborate Discussion , hence it is not just to sound better. Unnecessary redundant verbiage in Axioms & theorems can make it Strange & Confusing Eg "Positive Even Number greater than 0 , Divisible by 2 can be written as the Sum of 2 ODD Numbers , not Divisible by 2" is true though has a lot of unnecessary words ! – Prem Jun 30 '23 at 14:54
  • See this post. – Mauro ALLEGRANZA Jul 04 '23 at 07:40

1 Answers1

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I have 2 Interpretations of this Issue :

[ Interpretation A ] Prime-facie : It is unnecessary to state "Same type" , which is redundant & already agreed with , by convention.

It looks like "Over-Sight" by Tao.

Page 284 A.7 Equality :
Tao says that Equality is a relation linking 2 Objects of Same type.
Page 285 A.7 Equality :
Tao says that Set & Number should not be compared using/with/via Equality.
Like-wise , Number & Vector should not be compared using/with/via Equality.
More-over , the Convention is that when we use Equality on Different types , it is automatically false.
Tao says that it is "abuse of notation" to compare 2 Classes or 2 types : Eg $1=1/1$ & $1=1.0$ , though it is generally ok.

That is not a justification to make the Axiom use "Same type" , because , even with "abuse of notation" , we can still conclude that Eg $1/1=1$ & $1.0=1$ , giving Symmetry.

In other words , Symmetry Axiom using "Same type" is unnecessary & redundant : It is still true without that wording.

[ Interpretation B ] Basically : Symmetry Axiom is not concerned or bothered when we have 2 Classes or types. Symmetry Axiom makes no claim in that Scenario. It is outside the Scope of the Symmetry Axiom.

There is a Deeper Issue which Tao is not Explicitly mentioning. This is my thought , I am not claiming that Tao is agreeing with this. I get this interpretation when reading between the lines.

When using 2 Classes or types , all bets are off !
It may be Symmetric ; It may be Anti-Symmetric ; It may be something else !

Page 284 A.7 Equality :
Tao says : Depending on the Class or type , Equality Definition is made to suit that Case & It is arbitrary.
Equality Definition is generally not made on 2 Classes or 2 types.
When we have 2 Classes or 2 types , the Definition may or may not satisfy Symmetry : It is outside the Scope of the Symmetry Axiom.

[ Example 1 ] In Numerical Software terms , we may compare Integers & floating Point Numbers :
"1==1.0" may convert Integer "1" to floating Point "1.000000021" & then the comparison is not true.
"1.0==1" may convert floating Point "1.0" to Integer "1" & then the comparison is true.
No Symmetry.
[[ this is entirely Possible in Certain CPU Architectures ]]

Symmetry Axiom has nothing to say about that Case.

[ Example 2 ] Definition of $=$ on Individual Cases is arbitrary. Consider my new Definition of $=$ to mean "inclusion" or "is" :
Then "Socrates is (=) A human" is true , while "A human is (=) Socrates" is not true.
No Symmetry.
[[ this is entirely Possible in unorthodox theories ]]

Symmetry Axiom has nothing to say about that Case.

[ Example 3 ] Consider a fictitious Example.
Let "X" be some human & "the President" be the Post , where have 2 Classes or 2 types.
Thus , "X is (=) the President" may or may not be true while "The President is (=) X" may be always not true.
No Symmetry.
[[ this is entirely Possible in Object Oriented Programming ]]

Symmetry Axiom has nothing to say about that Case.

With that Interpretation : It is necessary to give the Criteria that we have "Same type" in Symmetry Axiom to exclude the Other Cases : Axiom makes no claim about the Other Cases.

Prem
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