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26 minutes ago, StrictlyLogical said:

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?

Okay. Any idea about content, e.g. more of this and less of that?

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1 hour ago, merjet said:

Okay. Any idea about content, e.g. more of this and less of that?

From my review of the contents Knapp's book seems a tad more advanced and "academic" than what a typical layperson might be interested in.  Of course all of this is pure speculation.

 

 

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7 hours ago, StrictlyLogical said:

what is [the mathematician ] presuming about the nature of x and about the nature of y when one writes

x+y

or

y+x

Mathematicians, and different mathematicians, mean different things depending on context. The context is either stated explicitly or reasonably gleaned per a given book or article.

So, just to narrow down, let's look at just two of the different contexts. (They are different but they support each other anyway.) To avoid getting too complicated for the purposes of brief posting, I'll give only a sketch, leaving out a lot of details, and not explain every concept (such as 'free variable') and taking some liberties with the notation and concepts, and for ease of reading, I won't always include quote marks to distinguish mention as opposed to use. (So this is not as accurate as a more authoritative treatment).

So two contexts:

(1) General, informal (or informal mixed with formal) discussion in mathematics about natural numbers.  (2) Formal first order Peano arithmetic [I'll just call it 'PA' here].

 

(1) In general mathematics, we might taken commutativity of addition to be obvious and thus a given. Or one might say:

 

"Okay, I'm going to state some truths about natural numbers from which I can prove a whole bunch of other truths, even though they're obvious anyway. The truths about addition I want to mention are:

0 added to any number is just that number. In symbols: x+0 = 0.

The sum of a number and the successor of another (or same) number is just the successor of the sum of the number and the other number. In symbols: x+Sy = S(x+y), or, put another way (where 'S' is defined as '+1'), x+(y+1) = (x+y)+1.

The induction rule. 

Now, with those three truths, one of the many truths I can prove, without assuming anything about natural numbers or what they are, other than those three truths, is the commutativity of addition. In whatever way you conceive the natural numbers, as long that conception includes those three truths I just mentioned, then the commutativity of addition is proven true."

 

Notice that we can't do this with the real numbers, because the induction rule does not work for the real numbers. So, for real numbers, we would take commutativity as an axiom (or in set theory, we would prove commutativity from the properties of the real numbers as they are set theoretically "constructed"). 

 

(2) PA, as a system, has a formal first order language, with the primitive logical symbols (including '='  as a logical symbol) and certain primitive non-logical symbols.

 

The logical symbols are:

Infinitely many variables: x, y, etc.

-> (interpreted as the material conditional)

~ (interpreted as negation)

and, from '->' and '~' we can define:

& (interpreted as conjunction)

v (interpreted as inclusive disjunction) 

A (so that, where P(x) is any formula with 'x' occurring free, AxP is always interpreted as "for all x, P(x)") 

and, from 'A' and '~' we can define:

E (so that ExP(x) is always interpreted as "there is an x such that P(x)")

 

The non-logical symbols are :

0

S

+

*

We define 

S(0) =1

S(1) = 2

etc. 

When the language is interpreted: '0' is assigned to a particular member of the domain of the interpretation; 'S' is assigned to a 1-place function (operation) on the domain, '+' and '*' are each assigned to 2-place functions on the domain. 

With the "intended" ("standard") interpretation: the domain is the set of natural numbers, '0' is assigned to the number zero, 'S' is assigned to the successor operation, and '+' and '*' are assigned to the addition and multiplication operations respectively. 

And, since '=' is a logical primitive, we assign it to the identity (equality) relation on the domain. 

So for any interpretation (such that each variable, in its role as a free variable, is assigned to some member of the domain):

x+y

is assigned to the value of the '+' operation applied to the ordered pair: <the assigned value of x, the assigned value of y>.

And 

x+y = y+x

holds in the interpretation if and only if the value of x+y is identical with (is equal to) the value of y+x.


So, to answer your question, in the syntax of the formal system itself, nothing is assumed as to what 'x' and 'y' stand for. But with a formal interpretation of the system, 'x', as a free variable stands for some member of the domain and 'y',  as a free variable, stands for some member of the domain. And with the standard interpretation, the domain is the set of natural numbers.

However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure:

AxAy x+y = y+x (abbreviated Axy x+y = y+x)

And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).

Edited by GrandMinnow

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34 minutes ago, GrandMinnow said:

However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure:

AxAy x+y = y+x (abbreviated Axy x=y = y+x)

And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).

AxAy x+y = y+x (abbreviated Axy x+y = y+x) ?

Question if the equal should be a plus (shown red).

Edited by dream_weaver

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6 hours ago, GrandMinnow said:

Mathematicians, and different mathematicians, mean different things depending on context. The context is either stated explicitly or reasonably gleaned per a given book or article.

So, just to narrow down, let's look at just two of the different contexts. (They are different but they support each other anyway.) To avoid getting too complicated for the purposes of brief posting, I'll give only a sketch, leaving out a lot of details, and not explain every concept (such as 'free variable') and taking some liberties with the notation and concepts, and for ease of reading, I won't always include quote marks to distinguish mention as opposed to use. (So this is not as accurate as a more authoritative treatment).

So two contexts:

(1) General, informal (or informal mixed with formal) discussion in mathematics about natural numbers.  (2) Formal first order Peano arithmetic [I'll just call it 'PA' here].

 

(1) In general mathematics, we might taken commutativity of addition to be obvious and thus a given. Or one might say:

 

"Okay, I'm going to state some truths about natural numbers from which I can prove a whole bunch of other truths, even though they're obvious anyway. The truths about addition I want to mention are:

0 added to any number is just that number. In symbols: x+0 = 0.

The sum of a number and the successor of another (or same) number is just the successor of the sum of the number and the other number. In symbols: x+Sy = S(x+y), or, put another way (where 'S' is defined as '+1'), x+(y+1) = (x+y)+1.

The induction rule. 

Now, with those three truths, one of the many truths I can prove, without assuming anything about natural numbers or what they are, other than those three truths, is the commutativity of addition. In whatever way you conceive the natural numbers, as long that conception includes those three truths I just mentioned, then the commutativity of addition is proven true."

 

Notice that we can't do this with the real numbers, because the induction rule does not work for the real numbers. So, for real numbers, we would take commutativity as an axiom (or in set theory, we would prove commutativity from the properties of the real numbers as they are set theoretically "constructed"). 

 

(2) PA, as a system, has a formal first order language, with the primitive logical symbols (including '='  as a logical symbol) and certain primitive non-logical symbols.

 

The logical symbols are:

Infinitely many variables: x, y, etc.

-> (interpreted as the material conditional)

~ (interpreted as negation)

and, from '->' and '~' we can define:

& (interpreted as conjunction)

v (interpreted as inclusive disjunction) 

A (so that, where P(x) is any formula with 'x' occurring free, AxP is always interpreted as "for all x, P(x)") 

and, from 'A' and '~' we can define:

E (so that ExP(x) is always interpreted as "there is an x such that P(x)")

 

The non-logical symbols are :

0

S

+

*

We define 

S(0) =1

S(1) = 2

etc. 

When the language is interpreted: '0' is assigned to a particular member of the domain of the interpretation; 'S' is assigned to a 1-place function (operation) on the domain, '+' and '*' are each assigned to 2-place functions on the domain. 

With the "intended" ("standard") interpretation: the domain is the set of natural numbers, '0' is assigned to the number zero, 'S' is assigned to the successor operation, and '+' and '*' are assigned to the addition and multiplication operations respectively. 

And, since '=' is a logical primitive, we assign it to the identity (equality) relation on the domain. 

So for any interpretation (such that each variable, in its role as a free variable, is assigned to some member of the domain):

x+y

is assigned to the value of the '+' operation applied to the ordered pair: <the assigned value of x, the assigned value of y>.

And 

x+y = y+x

holds in the interpretation if and only if the value of x+y is identical with (is equal to) the value of y+x.


So, to answer your question, in the syntax of the formal system itself, nothing is assumed as to what 'x' and 'y' stand for. But with a formal interpretation of the system, 'x', as a free variable stands for some member of the domain and 'y',  as a free variable, stands for some member of the domain. And with the standard interpretation, the domain is the set of natural numbers.

However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure:

AxAy x+y = y+x (abbreviated Axy x+y = y+x)

And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).

I studied set theory in university.  I studied group theory and quantum field theory for masters.  I’ve studied chaos theory and fractal dimension in my spare time and  I’ve read the Emperor’s New Mind, Metamagical Themas, Godel Escher Bach... why is it I have the deepest conviction that although most of these are interesting and useful they are no where nearly as profound and real an intellectual achievement as grasping Objectivism... many years later?

I have great respect for so much of what iI learned in academia and I did quite well but I truly am of the belief, and sometimes it shocks me to think it... after a BSc, and an MSc,  (and a professional degree which I will not divulge) ... after all of that... I still did not know how to truly think snd know until Rand and Peikoff.

 

I hate to say it but when I hear of successor functions and when I browse a chapter entitled “Building the real numbers” ... from my old set theory text... I can’t help but think something is wrong... and wonder what mathematics could have become if based on Objective philosophy.

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21 hours ago, merjet said:

Any idea about content, e.g. more of this and less of that?

I finished reading Knapp’s book, Mathematics is About the World.

I rate it 5 stars, but with some room for improvement.

Knapp barely mentions arithmetic and counting. More about arithmetic would strengthen his thesis that mathematics is about the world. The positive integers used for counting (and zero) form the foundation for the real numbers. Understanding addition and subtraction of fractions call upon the important concepts of unit and transformation, which he does use extensively for different topics – measuring and vector spaces.

As an aside, as I have already indicated, mathematics is also about the way we think about the world. Mathematicians “extrapolate” concepts beyond perceptual reality. Examples are complex numbers and matrices with more than 3 dimensions.

Edited by merjet

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7 hours ago, GrandMinnow said:

What’s wrong with the successor function? It’s basic arithmetic. It’s basic to the computer programs that are running your computer. I don’t know what your objection is. 

I don't object to the function as such, but its use to "define" or "build" the natural numbers and the philosophy underlying that use.

 

Number and quantity are aspects of things in reality.  We identify and conceive of them ultimately by perceiving things which are, and we do so typically at a very young age.

Years after leaning what number and quantity mean, what whole numbers of things are, and knowing how to count things and repeated actions, add, subtract etc. for some reason some of us find it necessary to replace the action or exercise (real or mental) of adding 1, with a successor function, and dispense with the concept of number we already know from reality, by engaging in an exercise of "generating" the natural numbers by repeated "application" of a function.

This exercise does not create the natural numbers nor does it in any way conceptually define for us what whole numbers are... it creates a parade of symbols that stand for the things we already know.

 

It's not as bad as "building" numbers from artificial and imagined things (sets or power sets of the "null set" and sets of those sets), which mimic the action of counting, or accumulating real objects through a recursive exercise, but is performed with units of fantasy rather than of reality... all they are doing is essentially counting the repeated application of their recursions on fictional things, which symbolically simulates incremental accumulation (which algorithm implicitly relies on the mathematician's antecedent concept of number to verify all along that his model is appropriate for his purpose). But why choose to "build" numbers from fantasy... when we can ground numbers in reality, and take it from there?

 

it is as though, a mathematician earnest to learn about and explore number and quantity, must do his very best to ignore or distance himself from reality, that the more remote and disconnected from the things which actually exhibit number and quantity, the better he will understand number and quantity.

 

I think works like Knapp's and Binswanger's are desperately needed.

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37 minutes ago, merjet said:

I finished reading Knapp’s book, Mathematics is About the World.

I rate it 5 stars, but with some room for improvement.

Knapp barely mentions arithmetic and counting. More about arithmetic would strengthen his thesis that mathematics is about the world. The positive integers (and zero) form the foundation for the real numbers. Understanding addition and subtraction of fractions call upon the important concepts of unit and transformation, which he does use extensively for different topics – measuring and vector spaces.

As an aside, as I have already indicated, mathematics is also about the way we think about the world. Mathematicians “extrapolate” concepts beyond perceptual reality. Examples are complex numbers and matrices with more than 3 dimensions.

That was fast.  Well now I have to get back at reading it then...  5 stars is impressive!

 

I agree.

But I would qualify that an extrapolation about knowledge about the world is ultimately still about the world.  "Extrapolation" cannot mean wholly disconnecting from perceptual reality.. extrapolation implies a continuity or connection with the world because it applies to our knowledge.  We can only have knowledge about that which is... fantasies about that which is not do not count as knowledge, and any extrapolation of any fantasy is just more fantasy.  

Certainly math includes concepts which are not directly observed in reality... and yes Math includes concepts about the way we think and compute, but it must relate back to the things about which we think, the quantities and number of things in reality.

Here I am not saying that every exercise in Mathematics IS about the world, no more than every scientific theory or every philosophy in history was about the world (many were NOT), but that VALID Mathematics is ultimately grounded in reality and hence is about the world.

 

Complex numbers are useful for computation of various quantities.  Matrices traditionally are useful in relation to linear algebra and solving equations, but I am not aware of the status of matrices with more than 3 dimensions. IMHO that would depend both on how well they are related to other math which is ultimately properly grounded and how useful they are to solve equations or computations.

EDIT:  Just reread a previous post of yours.  Note, a 2 dimensional matrix can be used to solve N unknowns in N equations.  A matrix with 3 or more dimensions is quite a different kind of thing. 

 

Edited by StrictlyLogical

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45 minutes ago, StrictlyLogical said:

EDIT:  Just reread a previous post of yours.  Note, a 2 dimensional matrix can be used to solve N unknowns in N equations.  A matrix with 3 or more dimensions is quite a different kind of thing. 

In one sense the dimension of a matrix is always 2 -- it has rows and it has columns. The usual meaning of a matrix's dimension is the number of rows and the number of columns -- e.g. 2x2, 3x3, 4x4, ... mxn (link). My saying "matrices with more than 3 dimensions" was less than exact. My intent was a 4x4, 5x5, etc. which do not have spatial counterparts. I don't know what you mean by "matrix with 3 or more dimensions."

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On 7/30/2019 at 7:03 AM, 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?

He will be making a "quasi-book" out of his HBL posts on philosophy of mathematics since 1998, with an overview essay. He wrote a post announcing the book on HBL a couple of weeks ago. The theme will be that Plato and Kant need to be expelled from philosophy of mathematics.

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2 hours ago, merjet said:

In one sense the dimension of a matrix is always 2 -- it has rows and it has columns. The usual meaning of a matrix's dimension is the number of rows and the number of columns -- e.g. 2x2, 3x3, 4x4, ... mxn (link). My saying "matrices with more than 3 dimensions" was less than exact. My intent was a 4x4, 5x5, etc. which do not have spatial counterparts. I don't know what you mean by "matrix with 3 or more dimensions."

A two dimensional matric has elements which are indexed in two dimensions

Aij

where "i" is the index in one dimension and "j" is the index in the other dimension.  The total number of elements is mxn for i = 1, ... m, and j = 1, ..., n

A matrix of 3 or more dimensions has elements indexed in 3 or more dimensions

Aijk

for example.

Here the "rules" for matrix operations would need to be generalized.

As in

http://www.iaeng.org/publication/WCE2010/WCE2010_pp1824-1828.pdf 

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6 hours ago, StrictlyLogical said:

I don't object to the function as such, but its use to "define" or "build" the natural numbers and the philosophy underlying that use.

[...]

for some reason some of us find it necessary to replace the action or exercise (real or mental) of adding 1, with a successor function, and dispense with the concept of number we already know from reality, by engaging in an exercise of "generating" the natural numbers by repeated "application" of a function.

No one dispenses with the ordinary notion of natural numbers. But regarding in terms of a function allows mathematics to characterize and handle the natural numbers in an exact way, and without having to say each time "what we do when we start with zero and then add one, and keep adding one after another",  not just from the outset (such as the PA axioms) but when the study gets much more complicated. The use of the notion of the successor function is utterly basic even to the study of computablity that has enabled the advent of such things as the computer you are using now. 

Also note: The successor function is not used to build or define the natural numbers with PA. And in set theory, there are equivalent ways of defining 'is a natural number', some of them not using a successor operation. Then there is also proving that there is the set whose members are all and only the natural numbers, and that proof "basically" amounts to pointing out that there is the set that has 0 and then the next after 0, and so on, and no other members*. And, to me, that seems pretty close to the common, informal, everyday sense too. 

* Though, to be fair, this is accomplished by using "surrogate" notions in terms of sets.

Quote

[...] it is as though, a mathematician earnest to learn about and explore number and quantity, must do his very best to ignore or distance himself from reality, that the more remote and disconnected from the things which actually exhibit number and quantity, the better he will understand number and quantity.

It would be a rare mathematician or philosopher of mathematics who would advocate anything that extreme or even the essence of it. On the contrary. Meanwhile, mathematicians have found that "idealizations" in mathematics are useful (or perhaps even essential) for developing the mathematics for the sciences. And formalization offers the ultimate objectivity in settling any question whether a purported mathematical proof (when put formally) is indeed a proof, as formalization entails that there is an algorithm to determine of any purported proof (when put formally) whether it is indeed a proof. Some mathematicians find this to be useful, or philosophically welcome, or just interesting. Other mathematicians don't care about it. And, of course, any non-mathematician is free to disregard it or disdain it. 

Edited by GrandMinnow

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