Kevin’s Reviews > Introduction to Mathematical Philosophy > Status Update
Kevin
is on page 77 of 224
Ch7:Rational, Real, and Complex Numbers;
With the definition of relations in hand, the theory of number types can be examined. The real rational numbers can be constructed with any ratio of the inductive numbers defined in chapters 2 and 3. Complex numbers arise in a similar, though not obvious, manner. This discussion seems to allude to the notion of algebraic numbers, but Russell doesn't call them algebraic.
— Jan 18, 2015 10:20PM
With the definition of relations in hand, the theory of number types can be examined. The real rational numbers can be constructed with any ratio of the inductive numbers defined in chapters 2 and 3. Complex numbers arise in a similar, though not obvious, manner. This discussion seems to allude to the notion of algebraic numbers, but Russell doesn't call them algebraic.
Like flag
Kevin’s Previous Updates
Kevin
is on page 131 of 224
Ch12:Selections and the Multiplicative Axiom;
This chapter is quite difficult. It builds towards the uncertainty surrounding whether an infinite number can be "multipliable". It starts with the assumption that given two sets α and β with terms μ and ν, μ*ν is defined as the number of ordered couples formed by picking the first coupled term from α and the second from β. This is the "Multiplicative Axiom".
— Jan 21, 2015 09:34AM
This chapter is quite difficult. It builds towards the uncertainty surrounding whether an infinite number can be "multipliable". It starts with the assumption that given two sets α and β with terms μ and ν, μ*ν is defined as the number of ordered couples formed by picking the first coupled term from α and the second from β. This is the "Multiplicative Axiom".
Kevin
is on page 117 of 224
Ch11:Limits and Continuity of Functions;
This chapter is essentially the epsilon-delta definition of a limit introduced in Calculus I courses. The definition of continuity proceeds: A function takes an argument and produces a value. If, at every point within an interval, the function value has a limit which agrees when the argument is approached from "above or below", then the function is continuous on the interval.
— Jan 21, 2015 07:16AM
This chapter is essentially the epsilon-delta definition of a limit introduced in Calculus I courses. The definition of continuity proceeds: A function takes an argument and produces a value. If, at every point within an interval, the function value has a limit which agrees when the argument is approached from "above or below", then the function is continuous on the interval.
Kevin
is on page 107 of 224
Ch10:Limits and Continuity;
The definition of continuity given by Cantor and Dedekind is closer to the "absence of separateness" than the presence of some other properties. The naturals and rationals have separation between terms while the series of all reals does not. Also, the concept of a limit is not quantitative, but ordinal.(If there exists a finite number that cannot be succeeded, then an upper limit exists.)
— Jan 20, 2015 09:04PM
The definition of continuity given by Cantor and Dedekind is closer to the "absence of separateness" than the presence of some other properties. The naturals and rationals have separation between terms while the series of all reals does not. Also, the concept of a limit is not quantitative, but ordinal.(If there exists a finite number that cannot be succeeded, then an upper limit exists.)
Kevin
is on page 97 of 224
Ch9:Infinite Series and Ordinals;
In addition to cardinal numbers that describe the size of a progression, there are ordinal numbers describing the order type of a well-ordered series("one in which every sub-class has a first term.") "An ordinal number means the relation-number of a well-ordered series." Ordinal and cardinal numbers obey their own set of arithmetic laws (commutative, associative,...)
— Jan 20, 2015 10:53AM
In addition to cardinal numbers that describe the size of a progression, there are ordinal numbers describing the order type of a well-ordered series("one in which every sub-class has a first term.") "An ordinal number means the relation-number of a well-ordered series." Ordinal and cardinal numbers obey their own set of arithmetic laws (commutative, associative,...)
Kevin
is on page 89 of 224
Ch8:Infinite Cardinal Numbers;
It's not absolute that infinite collections exist. To assume so is the "axiom of infinity". But, there is "no logical reason against infinite collections" as mathematical induction shows that for any natural number n, one can choose n+1. Assigning a number to the size of the collection is called its cardinality. The natural numbers has cardinality Aleph_0 ;and all reals is 2^(Aleph_0)
— Jan 20, 2015 08:48AM
It's not absolute that infinite collections exist. To assume so is the "axiom of infinity". But, there is "no logical reason against infinite collections" as mathematical induction shows that for any natural number n, one can choose n+1. Assigning a number to the size of the collection is called its cardinality. The natural numbers has cardinality Aleph_0 ;and all reals is 2^(Aleph_0)
Kevin
is on page 63 of 224
Ch6:Similarity of Relations;
"...what matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations."
When building definitions for our basis of mathematics, we must look for "likeness" (one-one relations and their correlates). Likeness, or "similarity", gives rise to structure. If I'm not mistaken, these terms are akin to isomorphs in modern parlance.
— Jan 18, 2015 08:44PM
"...what matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations."
When building definitions for our basis of mathematics, we must look for "likeness" (one-one relations and their correlates). Likeness, or "similarity", gives rise to structure. If I'm not mistaken, these terms are akin to isomorphs in modern parlance.
Kevin
is on page 52 of 224
Ch4:Kinds of Relations;
All relations can be derived from "one-many" relations, i.e. "if x has relation in question to y, there is no other term x' which also has the relation to y."
One-one relations can be seen by conjunction of this with the converse: if y has relation to x, there is no y' with same relation to x.
Some definitions:"Referent" is the term from the relation; "relatum" is the term to which it goes.
— Jan 18, 2015 07:03PM
All relations can be derived from "one-many" relations, i.e. "if x has relation in question to y, there is no other term x' which also has the relation to y."
One-one relations can be seen by conjunction of this with the converse: if y has relation to x, there is no y' with same relation to x.
Some definitions:"Referent" is the term from the relation; "relatum" is the term to which it goes.
Kevin
is on page 42 of 224
Ch3: The Definition of Order;
"The order lies, not in the class of terms, but in a relation among the members of the class."
— Jan 17, 2015 07:29PM
"The order lies, not in the class of terms, but in a relation among the members of the class."
