Seeing that the price of Pat Corvini's two sets of lectures on number:

1. Two, Three, Four and All That; and

2. Two, Three, Four and All That: The Sequel

were reduced to $2.75 each, I bought and listened to them. Each set contains 3 lectures. Link.

The main topic is her view of numbers. A lesser topic is criticism of Cantor's claims about infinite sets, and his method, with she calls postulational and contructive. Corvini does not say so, but the postulational/contructive philosophical view is epitomized by the famous mathematician David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.

Her method focuses on "the what" of numbers, whereas Cantor's methods focuses on "the how" of numbers. She sharply distinguishes between counting -- which use only the positive integers -- and measuring -- whose domain is the real numbers (integers + rationals + irrationals). Cantor's method of one-to-one correspondence blurs the distinction.

She talks about Cantor in the 1st and 3rd lectures of The Sequel. The last 1/3rd or so of Sequel #2 and the first half or so of Sequel #3 elaborate her view of measurement. Then she returns to Cantor and the postulational/constructive view of the rational and irrrational numbers. In her view there are two sorts of infinities -- counting (conceptualized by the positive integers) and measuring (conceptualized by real numbers and attained by subdividing). The postulational/constructive method blurs the distinction and treats open-ended construction like a concrete.

I much agree with what she says, but believe there are even stronger criticisms of Cantor's nonsense, ones that I have given on OL, especially in one thread starting about here. At one point in Sequel #1, Corvini talks in terms of 2-to-1 correspondence, but not any wider range of multiple-to-1 correspondences. Nor does she utilize part-whole logic to criticize Cantor's nonsense.

What has she published in the professional journals. I could not find a reference to her publications anywhere.

This is her doctoral dissertation pertaining to some kind of a dia-electric circuit or substance. She is an engineer so she should know some mathematics.

This expands on the analysis of rational versus irrational numbers on the postuational/constructive view ("P-view") versus Corvini's (and my) objective view ("O-view"). As indicated above, Corvini says the P-view emphasizes "the how" and the O-view "the what." Also, the P-view makes a strong distinction between the rational and irrational numbers. Per Cantor the former are countably infinite (a 1-to-1 correspondence is possible) but the latter are not (a 1-to-1 correspondence is impossible). The O-view makes a strong distinction between the integers and non-integers (rational and irrational numbers).

On the P-view an irrational number is completely defined by a Dedekind cut. On the O-view an irrational number is defined as the limit of an ordered sequence of rational numbers and can refer to an actual measurement, e.g. the length of the hypotenuse of a right triangle with the other two sides equal 1 (square root of 2) or the circumference of a circle with diameter of length 1 (pi).

The decimal expansion of a rational number is precise and has a pattern, e.g. 21/100 = 0.21 (only zeros follow),1/3 = 0.33333... (only 3's follow), and 1/7 = 0.142857142857... (the pattern repeats). The decimal expansion of an irrational number can only be approximate and there is no pattern to the digits. For example, the square root of 2 is 1.41421356237309504880168..... and pi is 3.14159265358979323846264.....

The square root of 2 is the limit of the sequence of rational numbers: 1.41, 1.414, 1.4142, 1.41421, 1.414213, 1.4142135,etc. Pi is the limit of the sequence of rational numbers: 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, etc.

Corvini's lectures do not address decimal expansion, but I am confident she would approve.

Measurement is not important on the P-view but is very important on the O-view.

Jesus H Kryst. What is the issue with rational and irrational numbers. Either a non integral real number is the ratio of two integers (which are relatively prime to each other ) or it isn't.

Do you see a "philosophical" problem here? I don't.

A real number is either rational or irrational. 3/5 is rational and the square root of two is not, for example.

Of the irrational number, either that number is the solution of a polynomial with integer coefficients or it is not.

Among irrational numbers the square root of two is the root of a polynomial with integer coefficients and pi is not (for example).

Objectivists should stay away from mathematics. In general they are particularly not equipped to deal with math.

Our Scholar in Residence Stephen is an exception. He knows his shit. Most Objectivist do not know dickey-do about math.

They know just about as much as did Ayn Rand, which is very little.

Our Scholar in Residence Stephen is an exception. He knows his shit.

Perhaps Stephen will tell us whether or not he accepts Cantor's transfinite nonsense that you so heartily endorse.

I do not endorse it heartily. But getting rid of the Axiom of Choice is like clipping the wings of an eagle.

A great deal of -useful- math (I mean math with applications in engineering and physics) flows from set theory. Try doing topology or the theory of differential manifolds (on which general relativity is based) without set theory. Good luck!

By the way your "nonsense" remark is proof positive of what I said about mathematics and Objectivists. You are a bigot and a mathematics ignoramus. You do not know fuck-all about that upon which you comment. Learn something --- then criticize.

A great deal of -useful- math (I mean math with applications in engineering and physics) flows from set theory. Try doing topology or the theory of differential manifolds (on which general relativity is based) without set theory. Good luck!

By the way your "nonsense" remark is proof positive of what I said about mathematics and Objectivists. You are a bigot and a mathematics ignoramus. You do not know fuck-all about that upon which you comment. Learn something --- then criticize.

Where did I say I rejected set theory? I only reject Cantor's nonsense.

A degree in math, several graduate courses in math, and a career as an actuary makes me a mathematics ignoramus? LOL. Learn something --- then criticize.

A great deal of -useful- math (I mean math with applications in engineering and physics) flows from set theory. Try doing topology or the theory of differential manifolds (on which general relativity is based) without set theory. Good luck!

By the way your "nonsense" remark is proof positive of what I said about mathematics and Objectivists. You are a bigot and a mathematics ignoramus. You do not know fuck-all about that upon which you comment. Learn something --- then criticize.

Where did I say I rejected set theory? I only reject Cantor's nonsense.

A degree in math, several graduate courses in math, and a career as an actuary makes me a mathematics ignoramus? LOL. Learn something --- then criticize.

Cantor's 'nonsense" is a logical and direct consequence of set theory. If you have set theory, the theory of transfinite cardinals and ordinals will follow logically, as does the night follow the day. You cannot have set theory without some version of the theory of cardinals and ordinals.

Cantor's 'nonsense" is a logical and direct consequence of set theory. If you have set theory, the theory of transfinite cardinals and ordinals will follow logically, as does the night follow the day.

So far as I have fathomed it, it seemed to me that Cantor's theory of transfinite numbers was sensible. I also liked the result that there was a way of acknowledging distinct orders of infinity, as in the transcendental numbers being an infinity larger than the infinity of irrational numbers. Years later I noticed that lack of a concept of orders of infinity in Nietzsche's mind was what allowed him to take perfectly seriously his argument for Eternal Recurrence. Whereas in truth, this day of this planet and our lives in this day will never recur, for the infinity of possible days of life, or possible fires in a fireplace for that matter, is some order of infinity above the order of infinity that would be an infinite time. Speaking of big numbers, congratulations to Bob on this his day 28,124 since birth! Hope this day is happy.

Stephen

~~~~~~~~~~~~~~~~

PS

That should have been "as in the transcendental irrational numbers being an infinity larger than the algebraic irrational numbers."

So far as I have fathomed it, it seemed to me that Cantor's theory of transfinite numbers was sensible. I also liked the result that there was a way of acknowledging distinct orders of infinity, as in the transcendental numbers being an infinity larger than the infinity of irrational numbers. Years later I noticed that lack of a concept of orders of infinity in Nietzsche's mind was what allowed him to take perfectly seriously his argument for Eternal Recurrence. Whereas in truth, this day of this planet and our lives in this day will never recur, for the infinity of possible days of life, or possible fires in a fireplace for that matter, is some order of infinity above the order of infinity that would be an infinite time. Speaking of big numbers, congratulations to Bob on this his day 28,124 since birth! Hope this day is happy.

Stephen

Marvelously happy, Stephen. And you helped to make my day.

Let me translate that 5 digit number into something most Objectivists can handle.

That is 77 years. Oy! Vey! What a ride!!! And its not over yet.

Years later I noticed that lack of a concept of orders of infinity in Nietzsche's mind was what allowed him to take perfectly seriously his argument for Eternal Recurrence. Whereas in truth, this day of this planet and our lives in this day will never recur, for the infinity of possible days of life, or possible fires in a fireplace for that matter, is some order of infinity above the order of infinity that would be an infinite time.

Not that I'm a supporter of any such BS but it makes you wonder if support for the Many-worlds interpretation of quantum mechanics is an attempt to get around the lesser

order of infinitely that would be infinite time. In the Many-worlds interpretation another universe is created at each branching point of every quantum interaction. This

branching point is not clearly defined so one could postulate an infinite number of branching points along each quantum collapse creating an infinite series of universes

of one, two, three, an infinite number of orders above infinite time alone.

See also the Many-minds interpretation which would also create many more universes.

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## BaalChatzaf

What has she published in the professional journals. I could not find a reference to her publications anywhere.

Got a hit!

http://siliconphotonics.ece.ucsb.edu/sites/default/files/publications/1995%20Current%20Injection%20in%20Semi-Insulating%20Indium%20Phosphide.PDF

This is her doctoral dissertation pertaining to some kind of a dia-electric circuit or substance. She is an engineer so she should know some mathematics.

Ba'al Chatzaf

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## merjet

AuthorThis expands on the analysis of rational versus irrational numbers on the postuational/constructive view ("P-view") versus Corvini's (and my) objective view ("O-view"). As indicated above, Corvini says the P-view emphasizes "the how" and the O-view "the what." Also, the P-view makes a strong distinction between the rational and irrational numbers. Per Cantor the former are countably infinite (a 1-to-1 correspondence is possible) but the latter are not (a 1-to-1 correspondence is impossible). The O-view makes a strong distinction between the integers and non-integers (rational and irrational numbers).

On the P-view an irrational number is

completely defined bya Dedekind cut. On the O-view an irrational number isdefinedas the limit of an ordered sequence of rational numbers and can refer to an actual measurement, e.g. the length of the hypotenuse of a right triangle with the other two sides equal 1 (square root of 2) or the circumference of a circle with diameter of length 1 (pi).The decimal expansion of a rational number is precise and has a pattern, e.g. 21/100 = 0.21 (only zeros follow),1/3 = 0.33333... (only 3's follow), and 1/7 = 0.142857142857... (the pattern repeats). The decimal expansion of an irrational number can only be approximate and there is no pattern to the digits. For example, the square root of 2 is 1.41421356237309504880168..... and pi is

3.14159265358979323846264.....

The square root of 2 is the limit of the sequence of rational numbers:

1.41, 1.414, 1.4142, 1.41421, 1.414213, 1.4142135,etc.

Pi is the limit of the sequence of rational numbers:

3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, etc.

Corvini's lectures do not address decimal expansion, but I am confident she would approve.

Measurement is not important on the P-view but is very important on the O-view.

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## BaalChatzaf

Jesus H Kryst. What is the issue with rational and irrational numbers. Either a non integral real number is the ratio of two integers (which are relatively prime to each other ) or it isn't.

Do you see a "philosophical" problem here? I don't.

A real number is either rational or irrational. 3/5 is rational and the square root of two is not, for example.

Of the irrational number, either that number is the solution of a polynomial with integer coefficients or it is not.

Among irrational numbers the square root of two is the root of a polynomial with integer coefficients and pi is not (for example).

Objectivists should stay away from mathematics. In general they are particularly not equipped to deal with math.

Our Scholar in Residence Stephen is an exception. He knows his shit. Most Objectivist do not know dickey-do about math.

They know just about as much as did Ayn Rand, which is very little.

Sorry for the rant, folks.

Ba'al Chatzaf

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## merjet

AuthorPerhaps Stephen will tell us whether or not he accepts Cantor's transfinite nonsense that you so passionately endorse.

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## BaalChatzaf

I do not endorse it heartily. But getting rid of the Axiom of Choice is like clipping the wings of an eagle.

A great deal of -useful- math (I mean math with applications in engineering and physics) flows from set theory. Try doing topology or the theory of differential manifolds (on which general relativity is based) without set theory. Good luck!

By the way your "nonsense" remark is proof positive of what I said about mathematics and Objectivists. You are a bigot and a mathematics ignoramus. You do not know fuck-all about that upon which you comment. Learn something --- then criticize.

Ba'al Chatzaf

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## merjet

AuthorWhere did I say I rejected set theory? I only reject Cantor's nonsense.

A degree in math, several graduate courses in math, and a career as an actuary makes me a mathematics ignoramus? LOL. Learn something --- then criticize.

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## BaalChatzaf

Cantor's 'nonsense" is a logical and direct consequence of set theory. If you have set theory, the theory of transfinite cardinals and ordinals will follow logically, as does the night follow the day. You cannot have set theory without some version of the theory of cardinals and ordinals.

Ba'al Chatzaf

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## merjet

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## Guyau

So far as I have fathomed it, it seemed to me that Cantor's theory of transfinite numbers was sensible. I also liked the result that there was a way of acknowledging distinct orders of infinity, as in the transcendental numbers being an infinity larger than the infinity of irrational numbers. Years later I noticed that lack of a concept of orders of infinity in Nietzsche's mind was what allowed him to take perfectly seriously his argument for Eternal Recurrence. Whereas in truth, this day of this planet and our lives in this day will never recur, for the infinity of possible days of life, or possible fires in a fireplace for that matter, is some order of infinity above the order of infinity that would be an infinite time. Speaking of big numbers, congratulations to Bob on this his day 28,124 since birth! Hope this day is happy.

Stephen

~~~~~~~~~~~~~~~~

PS

That should have been "as in the transcendental irrational numbers being an infinity larger than the algebraic irrational numbers."

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## BaalChatzaf

Marvelously happy, Stephen. And you helped to make my day.

Let me translate that 5 digit number into something most Objectivists can handle.

That is 77 years. Oy! Vey! What a ride!!! And its not over yet.

Ba'al Chatza

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## dennislmay

http://en.wikipedia.org/wiki/Many-worlds_interpretation

Not that I'm a supporter of any such BS but it makes you wonder if support for the Many-worlds interpretation of quantum mechanics is an attempt to get around the lesser

order of infinitely that would be infinite time. In the Many-worlds interpretation another universe is created at each branching point of every quantum interaction. This

branching point is not clearly defined so one could postulate an infinite number of branching points along each quantum collapse creating an infinite series of universes

of one, two, three, an infinite number of orders above infinite time alone.

See also the Many-minds interpretation which would also create many more universes.

http://en.wikipedia.org/wiki/Many-minds_interpretation

Once you open up things to the arbitrary there is no confining the infinite orders that

can be postulated to result.

Dennis

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