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Math and reality

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20 minutes ago, StrictlyLogical said:

The expression "1+A" IS not the expression "A+1".

but what 1+A IS, is identical to what A+1 IS, given what 1+A means and what A+1 means.

How is this not a contradiction? How can the two be identical but not the same? I don't deny that the expressions are what they mean. I'm saying they mean different things. One plus 'a' means that you are adding an 'a' to a one, and 'a' plus one means you're adding a one to an 'a'. These are different expressions. And when you place them on opposite sides of an equals sign, they become part of an equation where 'a' must be the same number in each expression. But that doesn't change the fact that both expressions have different meanings on their own.

But just because two things are unequal in one aspect doesn't mean they can't be equal in another. The two expressions are equal in their numerical results. But you have to solve the equation first to get the results.

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There has been objection in the past to substituting 'A=A' for 'A is A', and validly so.

In math, 3 is 3 and 3=3 reduce to the same, because in number, every instance of 3 is exactly the same.

The meaning of 'A is A' is 'a thing is itself'. In number, the referent is an abstraction. The number stands in for the relationship of a group to one of its members taken as a unit.

Using the membership/relationship/group/unit notion, should make the transitive property of (a+b)=(b+a) seems like an exercise in mental gymnastics.

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1 hour ago, MisterSwig said:

How is this not a contradiction? How can the two be identical but not the same? I don't deny that the expressions are what they mean. I'm saying they mean different things. One plus 'a' means that you are adding an 'a' to a one, and 'a' plus one means you're adding a one to an 'a'. These are different expressions. And when you place them on opposite sides of an equals sign, they become part of an equation where 'a' must be the same number in each expression. But that doesn't change the fact that both expressions have different meanings on their own.

But just because two things are unequal in one aspect doesn't mean they can't be equal in another. The two expressions are equal in their numerical results. But you have to solve the equation first to get the results.

"The man sitting in the chair in my office"

is not the same sentence as

"The man who ate with my wife and son this morning"

The words, the expressions are different.

But an evaluation of what they mean, the person in reality to which they refer is the same person, me.

[A side effect of these particular expressions is that you will learn something different about the referent ... but that is not the point.]

 

The point is that the expression or the sentence is not the same as the referent or meaning of the expression or sentence.

In an equation, the "=" implicitly denotes a context of both sides of the equation "as evaluated"

 

As for adding 1 and a or adding a and 1, these are not different if "adding" means bringing together a and 1.  "+" is not here time ordered, although one could conceive of a kind of time ordered sum.

a+b = SUM(a,b) is the sum of a and b, i.e.  a+b designates the result ofa together with b

An algorithm such as "starting with b then adding a to it" is different from a static math operator... I do not think (IMHO) the standard "+" has any time ordering associated with it... i.e. the math is not of the form "starting" "then"

 

1+A=A+1

Is much like

(1 AND A) IS (A AND 1)

 

which requires no induction.

Edited by StrictlyLogical

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1 hour ago, StrictlyLogical said:

As for adding 1 and a or adding a and 1, these are not different if "adding" means bringing together a and 1.

Let's recall why I brought this up. It was posited that the equation (1+a=a+1) could not be verified, because we would need to check it against every possible number, which is impossible to do because infinity. I know that wasn't your specific argument, SL, but now you seem to be saying that these expressions qua expressions are equal. So why do we need to verify the equation by impossibly plugging in an infinite series of numbers, when you've verified it without plugging in a single one?

1 hour ago, StrictlyLogical said:

I do not think (IMHO) the standard "+" has any time ordering associated with it... i.e. the math is not of the form "starting" "then"

It's not the plus symbol that has the ordering, it's the ordering that has the ordering. It depends on how you place the numbers and variables. If ordering didn't matter, then you wouldn't need to solve the equation to verify it. But since there is a difference between the expressions, you need to solve it to confirm that they total to the same number.

Edited by MisterSwig

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46 minutes ago, MisterSwig said:

Let's recall why I brought this up. It was posited that the equation (1+a=a+1) could not be verified, because we would need to check it against every possible number, which is impossible to do because infinity. I know that wasn't your specific argument, SL, but now you seem to be saying that these expressions qua expressions are equal. So why do we need to verify the equation by impossibly plugging in an infinite series of numbers, when you've verified it without plugging in a single one?

It's not the plus symbol that has the ordering, it's the ordering that has the ordering. It depends on how you place the numbers and variables. If ordering didn't matter, then you wouldn't need to solve the equation to verify it. But since there is a difference between the expressions, you need to solve it to confirm that they total to the same number.

I disagree.

As I said:

it's more like (1 AND A) IS (A and 1)

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34 minutes ago, StrictlyLogical said:

Anyone want to explain, in view of what I’ve stated above, how induction would be necessary here?  And IF necessary how it would be possible?

It isn’t necessary -- at least in your view -- but it is possible.

Task: Prove (1 + a) = (a + 1) is true for all natural numbers.

Method: mathematical induction

Base case: a = 1. (1 + a) = (a + 1) is obviously true.

Inductive step:

Show that if P(k) holds, then also P(k + 1) holds.

(1 + k) = (k + 1)

(1 + k) + 1 = (k +1) + 1

(1 + (k + 1)) = ((k +1) + 1)

QED.

From the linked page: "Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. ... Proofs by mathematical induction are, in fact, examples of deductive reasoning." 

In other words, mathematical induction relies on a chain of deductions.

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1 hour ago, merjet said:

It isn’t necessary -- at least in your view -- but it is possible.

Task: Prove (1 + a) = (a + 1) is true for all natural numbers.

Method: mathematical induction

Base case: a = 1. (1 + a) = (a + 1) is obviously true.

Inductive step:

Show that if P(k) holds, then also P(k + 1) holds.

(1 + k) = (k + 1)

(1 + k) + 1 = (k +1) + 1

(1 + (k + 1)) = ((k +1) + 1)

QED.

From the linked page: "Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. ... Proofs by mathematical induction are, in fact, examples of deductive reasoning." 

In other words, mathematical induction relies on a chain of deductions.

Yes induction works nicely.

and yup, it isn't necessary in my view...

I'd like to know the reasons why it would be necessary.. in someone else's view.

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1 hour ago, StrictlyLogical said:

Yes induction works nicely.

and yup, it isn't necessary in my view...

I'd like to know the reasons why it would be necessary.. in someone else's view.

I can't give think of any good reason why somebody else would think it necessary for 1+a = a+1.  On the other hand, there are other P(n) that could be proven by mathematical induction where the truth of P(n) for all n is not so intuitively obvious. Problems 3-7 here are examples. In the case of 1+a = a+1, using mathematical induction is akin to computing the area of a circle using integral calculus instead of using the simple formula pi*r^2.

Edited by merjet

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8 hours ago, merjet said:

It isn’t necessary -- at least in your view -- but it is possible.

Task: Prove (1 + a) = (a + 1) is true for all natural numbers.

Method: mathematical induction

Base case: a = 1. (1 + a) = (a + 1) is obviously true.

Inductive step:

Show that if P(k) holds, then also P(k + 1) holds.

(1 + k) = (k + 1)

(1 + k) + 1 = (k +1) + 1

(1 + (k + 1)) = ((k +1) + 1)

QED.

From the linked page: "Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. ... Proofs by mathematical induction are, in fact, examples of deductive reasoning." 

In other words, mathematical induction relies on a chain of deductions.

There are two things I really like about your induction.

1.  When you set a=1 you are able say it is trivial 1+a=a+1 because 1+1=1+1 trivially (a thing is itself) and a bunch of 1s is just a bunch of 1s.

2.  changing the brackets (1+k)+1 to 1+(k+1) without batting an eye.

 

Consider what every natural number N IS. 

It is the sum of N identical 1s i.e.

 1+1+...[N of them in total]...+1 

that is what it IS even if the notation I have used to identify it is awkward.

 

So 

1+A=A+1

1+(1+1+....[A of them]...+1)=(1+1+...[A of them]...+1)+1

Given they are all 1s this is trivially true.

or the last step could be a shift in brackets to surround all the identical 1s

(1+1+....[A+1 of them]...+1)=(1+1+...[A+1 of them]...+1)

Or for the last step we can shift the brackets to the left on just the lhs and follow the same notation 

(1+1+...[A of them]...+1)+1 = (1+1+...[A of them]...+1)+1

Which results in identity as well.

 

Anyone else want to give reasons for why induction would be necessary here?

Edited by StrictlyLogical

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Some topics that have been mentioned:

(1) MATHEMATICAL INDUCTION on NATURAL NUMBERS

Induction [by 'induction' in such contexts, I mean mathematical induction] is ordinarily used in these contexts:

* Proofs from the axioms of Peano arithmetic [by 'Peano arithmetic' I mean first order Peano arithmetic] in which induction is an axiom. Induction is needed because there are many things you can't prove about natural numbers from the Peano axioms without the induction axiom.

* Proofs from the axioms of set theory in which there is the set of all and only the natural numbers and that set admits induction. Induction is used  because it is the induction property of the natural numbers that permits many of the proofs about natural numbers.

* Proofs historically before Peano arithmetic or set theory. (But such proofs can be put in Peano arithmetic or set theory retroactively.) 

* Proofs in general mathematics in instances where Peano arithmetic or set theory are not necessarily explicitly mentioned. (But said mathematics can be formulated in Peano arithmetic or set theory.)

And none of this stems from any supposed need to avoid "derailment" from infinite cardinals or ordinal addition.  


(2) USE/MENTION

There is a distinction between a) symbols, or sequences of symbols that are terms, to stand for objects or range as a variables over objects and b) the objects that are symbolized. 

Single quote marks indicate that a linguistic object - a symbol or sequence of symbols - is referred to. (Actually, more exactly, for sequences we would use a concatenation marker, but that is too pedantic for this discussion.)

'2' is a symbol (a linguistic object), it is not a number. However, 2 is a number. 

But this has really nothing to do with stating the commutativity of addition. 


(3) IDENTITY

x = y 

means 

x and y are the same object. 

So '=' stands for the identity (equality) relation. 

If T and S are terms, then 

T = S 

means that T and S both name the same object.

Equivalence was mentioned. The identity relation is an equivalence relation, but there are equivalence relations other than identity. But there is nothing gained in this discussion by mentioning a warning against confusion with equivalence relations. There is no mistaking that '=' stands for identity.  

 

(4) An article titled 'Infinity plus one' was linked to. The title of that article is misleading. In regards to cardinals, we don't use 'infinity' as a noun, but rather 'is infinite' as an adjective. (This is different from such things as "points of infinity" in the extended reals system, as such points don't refer to cardinality but rather to ordering.)

 

(5) This comment was posted: "It was posited that the equation (1+a=a+1) could not be verified, because we would need to check it against every possible number, which is impossible to do because infinity." Just to be clear, that is not necessarily my own view, but rather it was part of a brief explanation of Hilbert's views, and even in that regard, the statement needs important qualifications such as those I mentioned. 

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On 7/25/2019 at 9:43 AM, StrictlyLogical said:

1+3

is not "presently" different from "once calculated"

1+3 MEANS the result of adding 1 and 3.

(1+3)=(10-6)

Would you agree that "1+3" is not presently the same as "10-6"? The latter is presently an expression with two addends. The former presently has a minuend and a subtrahend. Neither expression contains identical numbers. Even when calculated there is a conceptual distinction. The latter results in a sum, the former in a difference. Only the resulting numbers, themselves, are the same: 4=4. And to achieve that equality, one must perform the calculations.

Can it be the case that in this example both sides are presently different, yet in (1+A)=(A+1) they are not? Or are "1+3" and "10-6" also the same because they refer to the same number?

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1+3 is the same as 10-6. It is a  number.

‘1+3’ is different from ‘10-6’. They refer to the same number, but they are different strings of symbols  

This is an example of the distinction between use and mention. 

Edited by GrandMinnow

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22 hours ago, GrandMinnow said:

This is an example of the distinction between use and mention. 

How so? I'm not using these expressions to refer to themselves. I'm saying that they are not that thing to which they refer. 1+3 is not its referent 4. The same applies with the use-mention distinction. If I use "apple" to refer to an actual apple on a tree, the word itself is not that apple on the tree. They are two different things. Think of 1+3 as the word and 4 as the apple. 1+3 is "apple" in English, and 10-6 is "apple" in Chinese.

苹果

I guess that's "apple" in Chinese. I used Google translate to cut and paste.

So if you slap an equals sign between the English word and the Chinese logograms, how do you know whether they actually mean the same thing? You have to translate one or the other until each side has the same English word or Chinese characters. If you know both languages, you could do this in your own mind. Otherwise, you could have a Chinese speaker draw you a picture, and you draw him a picture, then you have two pictures of an apple. Problem solved.

The same process is used for my math equation. If you know both addition and subtraction, you can do the calculations/translations in your head. Otherwise, you could have a subtraction speaker draw you a picture, and you draw the picture for the addition side.

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Yes, exactly, the word 'apple' is not an apple.

And '1+3' is not a number, it is a word (or term).

And 1+3 is not a word, it is a number, and it is the number 4.

John Goodman is a person, not a name. 'John Goodman' is a name, not a person.

1+3 is a number, not a name. '1+3' is a name, not a number.

Edited by GrandMinnow

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On 7/27/2019 at 1:27 PM, StrictlyLogical said:

I'd like to know the reasons why it would be necessary.. in someone else's view.

To show that order didn't matter. 

2 / 4 = 4 / 2

clearly that isn't true, because order matters for division.

1 + a = a + 1

for this to be true, order can't matter.

And on a more (somewhat)  concrete level, the orientation of molecules can determine what chemical something is. It seems like you mentioned ordering very briefly, and perhaps it is true that you wouldn't always need mathematical induction when you abstract away time-ordering. But this is still an instance where it would be necessary.

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We prove that order doesn't matter (that addition is commutative). Strictly Logical is asking why (mathematical) induction is needed to prove that addition is commutative.

So the answer to 'why is induction needed to prove the commutativity of addition?' is not 'to prove the commutativity of addition'. 

The answer is that the commutativity of addition can't be proven from the Peano axioms without the axiom of induction (or, in set theory it is the inductive property of the natural numbers that permits the proof of commutativity of addition).

The relevant axioms in Peano arithmetic ('S' stands for successor; and the formulas are tacitly taken to be the universal closures):

x+0 = x

x+Sy = S(x+y)

From that, you can't prove

x+y = y+x

You need also the axiom schema of induction:

For all formulas P,

If P(0) and (for all x, P(x) implies P(x+1)), then for all x, P(x).

------

Or one could take

x+y = y+x as an axiom.

But it is not needed to do that, since it is already provable from the other axioms.

Edited by GrandMinnow

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17 hours ago, GrandMinnow said:

We prove that order doesn't matter (that addition is commutative). Strictly Logical is asking why (mathematical) induction is needed to prove that addition is commutative.

So the answer to 'why is induction needed to prove the commutativity of addition?' is not 'to prove the commutativity of addition'. 

The answer is that the commutativity of addition can't be proven from the Peano axioms without the axiom of induction (or, in set theory it is the inductive property of the natural numbers that permits the proof of commutativity of addition).

The relevant axioms in Peano arithmetic ('S' stands for successor; and the formulas are tacitly taken to be the universal closures):

x+0 = x

x+Sy = S(x+y)

From that, you can't prove

x+y = y+x

You need also the axiom schema of induction:

For all formulas P,

If P(0) and (for all x, P(x) implies P(x+1)), then for all x, P(x).

------

Or one could take

x+y = y+x as an axiom.

But it is not needed to do that, since it is already provable from the other axioms.

I'm curious.

Prior to any induction, (and without taking x+y=y+x) as an axiom) what are x and y?

i.e. what is one presuming about the nature of x and about the nature of y when one writes

x+y

or

y+x

Equivalently, at the point prior to induction, what does the mathematician intend to mean when she writes x+y or y+x.

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1 hour ago, William O said:

It looks like Harry Binswanger has a new book on philosophy of mathematics in the works. That should be interesting.

Regarding Robert Knapp's book, the author acknowledges Binswanger's help, and Binswanger wrote a 5-star review of it for Amazon. So I'm curious if you have any clues about how a book by Binswanger would differ from Knapp's?

Edited by merjet

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4 minutes ago, merjet said:

Regarding Robert Knapp's book, the author acknowledges Binswanger's help, and Binswanger wrote a 5-star review of it for Amazon. So I'm curious if you have any clues about how a book by Binswanger would differ from Knapp's?

Perhaps he has a slightly different audience in mind?  More for a layperson who is philosophically inclined, rather than a technical or scientific person who is mathematically inclined?

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