I didn’t see a forum where I thought this post fits well. If the moderators want to move it to another forum , that’s okay. Anyway, I’ve been posting to this blog for a while, and believe some would find an interest in a couple recent ones.
Marconi #6 This is one of a series of 11 that I wrote while reading a biography of Guglielmo Marconi, the inventor of wireless technology and often credited with inventing the radio. The post refers to John Galt.
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Thanks!
Information and Investment #1
Information and Investment #2
Information and Investment #3
Information and Investment #4
Merlin,
Concerning the rationality and the explanatory virtues of Cantor’s extensions of finite arithmetic, and the prior failure of the part-whole approach by Bolzano, you might like to look into pp. 207–12 of Philip Kitcher’s The Nature of Mathematical Knowledge (if you’ve not done so already).
Thanks for the reference to Kitcher. I read that book many years ago, but not recently and before I wrote the blog posts. On page 211 Kitcher says:
“By contrast, because he cleaves to the intuitive idea that a set must be bigger than any of its proper subsets, Bolzano is unable to define even an order relation between infinite sets. The root of the problem is that, since he is forced to give up the thesis that the existence of one-to-one correspondence suffices for identity of cardinality, Bolzano has no way to compare infinite sets with different members. Second, Cantor’s work yields a new perspective on an old subject: we have recognized the importance of one-to-one correspondence to cardinality; we have appreciated the difference between cardinal and ordinal numbers; we have recognized the special features of the ordering of natural numbers. But we do not even need to go so far into transfinite arithmetic to receive explanatory dividends. Cantor’s initial results on the denumerability of the rationals and algebraic numbers, and the nondenumerability of the reals, provide us with a new understanding of the difference between the real numbers and the algebraic numbers.”
In my view Kitcher’s view is rather one-sided, favoring Cantor’s ideas over Bolzano’s. “Bolzano is unable to define even an order relation between infinite sets.” Why not? While the proper subset method is unable to give an order relation between all infinite sets, it is able to give an order relation between some infinite sets. An example of the former is the rationals in the interval [0,2] and the reals in the interval [0,1]. An example of the latter is the integers and reals.
It seems Kitcher values the denumerability/nondenumerability criteria much more than I do. According to Cantor, the rationals are denumerable, but the reals are not. On the other hand, comparing the rational numbers to the reals can also be done on the criteria of decimal expansions. We know that rational numbers have finite or recurring decimals expansions and irrational numbers have non-finite or non-recurring decimals.
Stephen, I’m sure you know this, but I will give examples for other readers who might not.
Rational number examples:
2/7 = 0.2857142857142857….. infinite, recurring
3/10 = 0.3 finite
77238/100000 = 0.77238 finite
Irrational number examples:
sqrt(2) =1.414213562373095….. infinite, nonrecurring
pi = 3.1415926535897932384…. infinite, nonrecurring
Starting with any rational number with a finite decimal expansion, one could generate an unlimited number of partly irrational numbers by appending digits randomly (nonrecurring) on the right side. I believe that is as sound or more sound than Cantor’s diagonal argument for real numbers (link).
P.S.
4 hours ago, merjet said:
finite or recurring
That should be finite and recurring.
4 hours ago, merjet said:
non-finite or non-recurring
That should be non-finite and non-recurring.
On 12/19/2018 at 11:40 AM, merjet said:
Thanks for the reference to Kitcher. I read that book many years ago, but not recently and before I wrote the blog posts. On page 211 Kitcher says:
“By contrast, because he cleaves to the intuitive idea that a set must be bigger than any of its proper subsets, Bolzano is unable to define even an order relation between infinite sets. The root of the problem is that, since he is forced to give up the thesis that the existence of one-to-one correspondence suffices for identity of cardinality, Bolzano has no way to compare infinite sets with different members. Second, Cantor’s work yields a new perspective on an old subject: we have recognized the importance of one-to-one correspondence to cardinality; we have appreciated the difference between cardinal and ordinal numbers; we have recognized the special features of the ordering of natural numbers. But we do not even need to go so far into transfinite arithmetic to receive explanatory dividends. Cantor’s initial results on the denumerability of the rationals and algebraic numbers, and the nondenumerability of the reals, provide us with a new understanding of the difference between the real numbers and the algebraic numbers.”
In my view Kitcher’s view is rather one-sided, favoring Cantor’s ideas over Bolzano’s. “Bolzano is unable to define even an order relation between infinite sets.” Why not? While the proper subset method is unable to give an order relation between all infinite sets, it is able to give an order relation between some infinite sets. An example of the former is the rationals in the interval [0,2] and the reals in the interval [0,1]. An example of the latter is the integers and reals.
It seems Kitcher values the denumerability/nondenumerability criteria much more than I do. According to Cantor, the rationals are denumerable, but the reals are not. On the other hand, comparing the rational numbers to the reals can also be done on the criteria of decimal expansions. We know that rational numbers have finite or recurring decimals expansions and irrational numbers have non-finite or non-recurring decimals.
Stephen, I’m sure you know this, but I will give examples for other readers who might not.
Rational number examples:
2/7 = 0.2857142857142857….. infinite, recurring
3/10 = 0.3 finite
77238/100000 = 0.77238 finite
Irrational number examples:
sqrt(2) =1.414213562373095….. infinite, nonrecurring
pi = 3.1415926535897932384…. infinite, nonrecurring
Starting with any rational number with a finite decimal expansion, one could generate an unlimited number of partly irrational numbers by appending digits randomly (nonrecurring) on the right side. I believe that is as sound or more sound than Cantor’s diagonal argument for real numbers (link).
2/7
expressed in an arbitrary “math writing” system, such as base-7 would be finite…
Does this say anything about actual quantities in reality or more about how we express them?
For example are curved lengths in reality somehow incommensurate (not technically correct but I cant think of a better word) with straight lengths or is it merely a consequence of “expression”
Sorry I’m being quite vague and reaching … it’s a sense I’ve had for quite some time now. The quantities Rational and irrational numbers designate are not different in kind but our attempts at designation (manner, base number system etc) do differ in effectiveness.
2 hours ago, StrictlyLogical said:
2/7
expressed in an arbitrary “math writing” system, such as base-7 would be finite…
Does this say anything about actual quantities in reality or more about how we express them?
For example are curved lengths in reality somehow incommensurate (not technically correct but I cant think of a better word) with straight lengths or is it merely a consequence of “expression”
Sorry I’m being quite vague and reaching … it’s a sense I’ve had for quite some time now. The quantities Rational and irrational numbers designate are not different in kind but our attempts at designation (manner, base number system etc) do differ in effectiveness.
Decimal expansions in base 7 would still include finite, infinite non-recurring, and infinite recurring instances.
If by “actual quantities in reality” you mean the result of a measurement, then the result is a matter of precision, i.e. how many significant digits.
I doubt there is any mathematician who would say that rational and irrational numbers designate are not different in kind.
25 minutes ago, merjet said:
I doubt there is any mathematician who would say that rational and irrational numbers designate are not different in kind.
Wouldn’t that depend on the mathematician?
Consider two intersecting squares which share two points as a common corner, the first square oriented at 45 degrees relative to the second square (as rotated about that point), the first square having a side spanning opposite corners of the second square… i. e. the first square has a side coincident with the “diagonal” of the second square.
Consider whether the length of a side of the first square differs in kind from the length of a side of the second square and whether a written magnitude expressing those lengths differs in kind. Observe we can arbitrarily assign a unit of length to be equal to the length of a side of the first square or arbitrarily assign a unit length to be equal to the length of a side of the second square.
Thoughts?
2 hours ago, StrictlyLogical said:
Wouldn’t that depend on the mathematician?
My perspective was whether or not a number was the ratio of two integers, or is equivalent to such a ratio. It either is or isn’t.
2 hours ago, StrictlyLogical said:
Consider two intersecting squares which share two points as a common corner, the first square oriented at 45 degrees relative to the second square (as rotated about that point), the first square having a side spanning opposite corners of the second square… i. e. the first square has a side coincident with the “diagonal” of the second square.
Consider whether the length of a side of the first square differs in kind from the length of a side of the second square and whether a written magnitude expressing those lengths differs in kind. Observe we can arbitrarily assign a unit of length to be equal to the length of a side of the first square or arbitrarily assign a unit length to be equal to the length of a side of the second square.
Thoughts?
The lengths of the sides of the two are not different in kind in that they are both real numbers, if that’s what you mean. And, of course, we can’t measure lengths with unlimited precision. Any measurement we express in digits is equivalent to a rational number, e.g. the diagonal of a 1-inch square is 1.41421356237 with 12-digit precision. 1.41421356237 = 141421356237/100000000000
3 hours ago, merjet said:
The lengths of the sides of the two are not different in kind in that they are both real numbers, if that’s what you mean. And, of course, we can’t measure lengths with unlimited precision. Any measurement we express in digits is equivalent to a rational number, e.g. the diagonal of a 1-inch square is 1.41421356237 with 12-digit precision. 1.41421356237 = 141421356237/100000000000
Sort of. The further point was that if one arbitrarily defines the first square as having sides of unit length, the length of a side of the second square is an irrational number whereas if one arbitrarily defines the second square as having sides of unit length, the length of a side of the first square is an irrational number.
My meanderings about incommensurables are just ponderings likely not valid … nice blog posts about Cantor.