Mathematics, Logic, Set Theory, and Objectivism Options

[PhilE.Stein]

Topic Description: A general discussion of Objectivist ideas of mathematics, especially regarding mathematical logic and set theory.

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I'd like to know more about Objectivist views of mathematics, especially regarding mathematical logic and set theory. I have a number of questions. I encourage anyone with insight to address any of these questions or to offer their thoughts irrespective of my questions or to ask additional questions. I hope for a productive discussion that will probe for strengths and weaknesses of classical mathematical logic and set theory, intuitionist mathematics, Objectivist proposals for mathematics, and other approaches to mathematics.

1. What is the state of the art in Objectivist philosophy of mathematics? What Objectivist writers have made significant progress in this area, and what publications may one view?

2. What synopsis can be given of Objectivist views on mathematics? Has a systematic and rigorous statement of an Objectivist philosophy of mathematics been offered?

3. Are there Objectivist mathematicians who have published mathematics that embody particularly Objectivist mathematical concerns, approaches, or philosophy? Are there proposals for new systems of mathematical logic or for new axioms for set theory or mathematics? Or do Objectivist mathematicians find already existing systems and axioms satisfactory so that Objectivist alternatives are not needed?

4. How would mathematicians implement Objectivist ideas about mathematics? If the ideas are not suggested for implementation in mathematical theories, then in what way is it hoped that the ideas will affect mathematical practice?

5 How do Objectivists view mathematical logic and set theory? Is there a preference for classical mathematical logic, for intuitionist mathematical logic, or for other systems, approaches, or philosophies? Most particularly, if there is an Objectivist critique of classical mathematical logic and set theory, what is the critique and what alternatives do Objectivists offer?

6. Probably the most common definition of 'finite' and 'infinite' among mathematicians these days is:

x is finite if and only if x is equinumerous with a natural number.
x is infinite if and only if x is not finite.
(Note: The subject of the empty set had been raised in another thread. The empty set is finite, according the above definitions, since the empty set is a natural number.)

Also, set theories such as Zermelo-Fraenkel set theory, have an axiom that there exists a set w such that 0 is a member of w, and if n is a member of w, then the union of n and the singleton of n is a member of w.

What objections, if any, do Objectivists have to these formulations?

7. While constructivist and/or intuitionist mathematics might not be a perfect fit for Objectivism, these might be closer to Objectivism than mathematics set in classical mathematical logic. Though I know of Heyting arithmetic, I have not been able to find intuitionist axioms for other concerns, especially the real numbers or for set theory. What are some intuitionist systems for real numbers or for set theory and how do Objectivists regard these proposals?

8. Would non-standard analysis be preferred by Objectivists or does non-standard analysis carry assumptions that are more disagreeable to Objectivism than notions of infinity in standard analysis?

9. I've heard a little about mathematics that allows only finite sets (if I'm putting this correctly). How does such mathematics express the real numbers? How does such mathematics present its own meta-theory? Do such efforts offer alternatives for Objectivists?

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It is not required that in this thread we address all of these questions, but these questions do give us plenty to talk about. And I hope that all participants of this thread are allowed to comment or even digress upon anything that falls in the general area of mathematics or other subjects, such as epistemology or philosophy, that contribute to our understanding of mathematics. As well, I hope participants will be allowed to discuss the particulars of certain matters of mathematics as these may arise in the course of the larger subject.

Thanks in advance for whatever light anyone may shed upon this subject.

[Tom Rexton]

 
In this question and comments the term "order" is being used in two different ways, one as a simple progression and another as the mathematical concept of order.  As explained previously, the principle of induction is based on succession, not on "order", which is an inductive extension of succession.  There is, however, another formulation of mathematical induction that does make use of the concept order and which is possible once the concept of order is available:  In terms of properties, it is: if P(1) is true, and P(k) for all k < n -> P(n), then P(n) for all n.  This form can be proved using the original form and the concept of order. 

I understand now. My professor clarified that principle of induction is related to, but not logically dependent on, the well-ordering principle. In fact, strong induction can actually be used to prove the well-ordering principle.

 
More fundamental is the issue of mathematical axioms as such.  Mathematical axioms are not philosophical "self-evident" axioms.  The Peano "axioms" define a concept of a mathematical system <N, successor, 1>.  This system is a mathematical concept of method that requires a prior knowledge of the concepts and principles of number and arithmetic, without which no one could come up with it at all.  If someone did accidently stumble on it, it would be a meaningless game, or at best, a minor method of symbol manipulation pertaining to simple (non-numerical) progressions.  It would not be part of the science of mathematics, which without the concept of number would not exist beyond possibly simple geometry.  (The axioms do not define "number" as "that which satisfies ..." -- that what which satisfies ...?; there is no genus.)   
   
The "starting point" given by the axioms of the Peano system provides a basis from which to systematically define and derive the principles of arithmetic (ultimately all the way to the real and complex numbers, which historically was the goal): The axioms provide a means to organize and systematize mathematical knowledge, keeping track of logical dependencies of the principles, and helping to avoid circularity and other errors of reasoning with complex ideas.   
 
But while the theorems and definitions in the Landau development are based only on the axioms, it would be foolish to pretend that we could have no knowledge of them at all without the Peano axioms.  The axioms do not provide a starting point in a conceptual vacuum, somehow "presupposed" in advance of all knowledge of numbers and arithmetic.  The concepts of number and arithmetic are ultimately based on reference to reality in the form of man's cognitive ability to conceptualize units abstracted from reality and his ability to devise further concepts and principles of method to deal with them (see IOE).  That is what stops the "infinite regress" -- existence is the actual beginning.  The concepts "number", "succession" and "1" are given in the axiomatic formulation, so they cannot be defined in terms of it, but they are not literally "undefined" (see IOE) -- only not defined within the Peano system.  So don't confuse the developmental dependencies in the definitions and theorems about the Peano system with its conceptual dependencies on prior known concepts and principles of arithmetic that are implicit in the axioms.   

So as I understand it then, axioms are used only to organize a branch of mathematics into a deductive system. They are not literally the starting point in which mathematics developed both historically and in each individual's minds. That makes sense. After all, one does not learn mathematics by first starting with the axioms and then deducing everything one can! Neither did mathematics develop historically first with the axioms. On the contrary, the axioms came long after some branch(es) of mathematics had been highly developed. (For example, geometry in Euclid's Elements, wherein Euclid transformed geometry into an axiomatic system, rather than creating it purely from the axioms.)

 
Even Peano himself was explicit about this.  He did not have a theory of concepts to give a proper explanation of how, conceptually, the whole system is ultimately based on reality, but in his original 1889 The Principles of Arithmetic Presented by a New Method he said his system was about a method for numbers; he did not claim to provide a way to derive the laws of arithmetic through a contentless meaningless formalism, somehow miraculously popping out of conceptually "presupposed" axioms mentally levitating in an epistemological vacuum:   
So the axioms in the concept of the Peano system of course "depend" on concepts of number, which are then systematically derived from the axioms: All the principles derived in the Landau development are implicit in the axioms -- which were synthesized by Dedekind and Peano only after the basic logical connections were worked out in advance to inductively discover the proper axioms.  The axioms themselves had to be validated by showing how they in fact lead to the laws of arithmetic (see Thurston, chapter X, even though it contains some conceptual groping).  If these principles were not implicit in the axioms, the axioms would have been the wrong choice.     
   
The inductive formulation of the axioms and the deductions in the theory are reciprocal -- with the deductive science allowing for the discovery, understanding and proof of more complex principles than would otherwise be possible.  That is the sense in which the Peano system forms a "foundation" -- not as a "presupposed" starting point in a mental vacuum, with no cognitive dependencies at all.  Mathematics is a deductive science, but the inductive generalizations leading to the formulation and definitions are also part of the science.   
   
In particular, of course the axiom of mathematical induction depends on our concept of number and how we count (as well as on man's conceptual abilities of deductive and inductive logic).  What else are we talking about?  It is only with that knowledge that anyone could come up with the axiom at all, let alone know it is true.     
   
In this forum, we know better than to give Russell's answer, mentioned earlier, that [in mathematics] "we never know what we are talking about, nor whether what we are saying is true" (which claim is a consequence of his own inverted and free-floating conceptual hierarchies), but the cumulative impact of the epistemology behind how mathematics is routinely presented and taught is clear when even here the conceptual dependency of the Peano axioms is regarded as somehow suspiciously tainting the theory because they do not come from something either entirely arbitrary or otherwise independent from our understanding of numbers. 
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So mathematical concepts, definitions, conjectures, generalizations, axioms, etc. are formed inductively, then they are reorganized, systematized and proved deductively. Still, many mathematicians and philosophers alike tend to belittle the former (induction) and think that mathematics is all about the latter (deduction). They present mathematics as though it all starts with axioms and rules of deductive inference. I suppose this is due a lack of an explicit understanding of the nature of induction and its contribution to science, especially to mathematics.

[ewv]

ewv: x + y' = (x+y)', together with the starting point x+1=x', is the way you express the recursion. 
 
The use of nested parentheses, at this stage of the development, does not refer to groupings of three or more numbers added because you don't have addition yet -- you are in the process of defining it.  Nested parentheses with multiple groupings are used to explicitly represent the results, in a single closed formula, of some specific number of applications of the recursion using succession.  It illustrates, but does not by itself completely specify, the recursive process.

Fesapo: I think I have finally clarified what bothered me about this definition. I'll restate what I think to be the case and see if you can confirm/disconfirm my understanding.

This first part I have no problem with as a definition of "x+1":
x+1 = x'
This defines "addition of one" as succession.

And, then, chaining these togther we get:
1. x+1 = x'
2. x+1+1 = x''
3. x+3 = x'''
4. x+4 = x''''
. . .
. . .
y. x+y = x-y#of' [x with the y number of succesors applied]
y+1. x+y+1 = (x-y#of')' [another recursive application of (1)]
y+1. x+y' = (x+y)' [substituting the equality in (y) into (y+1)]

So, as written above, the "chaining" of cases leads to a definition where x+y is defined in terms of succession. This is then extended one case further to cover the case which defines x+y' in terms of succession applied to the summation.

I guess the problem I had was in thinking that the definition ought to end at the case y, rather than at y+1.

Fesapo: I do remember that in other demonstrations, such as with series, one specified them as follows:

1, 2, 3, 5, 8, 13, .... n, n+1

The pattern of the indices would usually be either
a_1, a_2, a_3,..., a_n for a finite sequence,
or something like
a_1, a_2, a_3,..., a_n, ...
or
a_1, a_2, a_3,..., a_n, a_n+1, ...
for an infinite sequence, but the 'n+1' in that case is optional, only used to emphasize a pattern.

For the Fibonacci sequence that you gave:

1, 2, 3, 5, 8, 13, .... n, n+1

you have mixed up the notation. You started with the actual terms, then switched to an index. You would express the nth term as
a_n, or as
a_n = a_n-2 + a_n-1 for n>2
not as its index "n". If it is terminated as a finite sequence then there would ordinarily be no point to specifying a term for n+1.

But that is all a matter of straightening out notation; it is not fundamental.

Fesapo: And the formulation in Landau follows that by offering the cases (1) and (n+1), or rather, (y+1).

The recursive formulation requires using y' to express the increment (succession) relating it to the previous term y as an expression intended to be repeatedly applied in the open-ended progression. You have to relate two terms in order to say anything. This is not a matter of optional notation. It is required for recursion.

Fesapo: What had troubled me about claiming that x+y' = (x+y)' was part of the definition of addition as a recursive function, is that it seemed to be an implicit definition of x+y.

It is an implicit function definition in that it specifies the function through a general relation. That is why it must by proved that this relation is in fact a function for all x and y, i.e., that the function exists and is single-valued, with a unique value for each x and y. (We discussed the uniqueness portion of this above in #66.)

You can't, at this stage of the development, assume that using the inductive principle in a recursive definition is a valid approach; You have to prove it. The axiom of induction is a rule of proof, which is not the same as definition, even though the axiom of induction and recursive definitions both use the same basic idea. You have to prove, using the axioms, that using the idea of repeated succession implicitly specified by a recursion does in fact define a function. Of course it all works out because recursion and the axiom of induction are so similar.

Fesapo: I still think that this is the case and that my version which defines x+y as "-with-y-number-of-successions" is a more directly defining one. But I suppose that one can properly say that x+y is defined such that those two endpoints are met, and this is also a valid definition.

A definition invoking "y number of successions to get to y" is not valid within the axiomatic framework at that stage of its development. You don't yet have the concept of indexing a sequence, or the concept of order to refer, in general, to all numbers up to a certain number used to reach that number. See the related discussions in Landau pp. ix-x in the preface, and in Thurston pp. 10-11.

You are giving a pre-axiomatic definition of addition based on counting. That knowledge is implicit in the formulation and eventually comes out of it, is required to get to the stage of the axiomatic formulation at all, and does not mysteriously disappear from knowledge once you begin analyzing the consequences of Peano's axioms. But you will not be able to proceed in the development without understanding the abstraction of the recursive definition, which is used in the inductive formulations and proofs in the subsequent development. Attempting to avoid the recursive definition is like counting on your fingers to avoid the more advanced abstract concepts and methods of addition and subtraction, or the temptation we all had at the beginning to short circuit Euclid by using principles that are "obvious" about triangles instead of following the progression of definitions and proofs using Euclid's axioms and theorems.

The recursive formulation is also a common abstraction used in computer programming, so you should already by familiar with it. For example, you recursively generate the Fibonacci sequence you gave above (each term is the sum of the two preceding terms) by initializing
a_1 = 1
a_2 = 2
and then computing
a_k = a_k-2 + a_k-1
as you increment k++ for k>2

More generally in computer programming, you can use recursive function calls in which a function calls itself (being careful that each instance has its own local variables). More theoretically, recursive functions play a crucial role in the theory of computability in computer science.

But that is all a different topic from the development in Landau. For present purposes you need the abstraction of recursive definitions to proceed with that development. If all you needed to do was simple calculations like balancing a check book, engineering analysis or quantum mechanics then the definition of addition presented in grade school would suffice and you wouldn't need the technical mathematical theory of numbers. But the goal here is to understand the conceptual nature and status of the mathematical theory of the number systems from simple counting up through real and complex numbers. To get to that you have to follow the concepts and theorems in Landau.

[ewv]

Tom Rexton: I understand now. My professor clarified that principle of induction is related to, but not logically dependent on, the well-ordering principle. In fact, strong induction can actually be used to prove the well-ordering principle.

Which in turn depends on simple ordering "<" (without specifying whether there is a least element). Induction precedes both simple ordering and well ordering, but depends on succession.

Tom Rexton: So as I understand it then, axioms are used only to organize a branch of mathematics into a deductive system. They are not literally the starting point in which mathematics developed both historically and in each individual's minds. That makes sense. After all, one does not learn mathematics by first starting with the axioms and then deducing everything one can! Neither did mathematics develop historically first with the axioms. On the contrary, the axioms came long after some branch(es) of mathematics had been highly developed. (For example, geometry in Euclid's Elements, wherein Euclid transformed geometry into an axiomatic system, rather than creating it purely from the axioms.)

Yes. Where else could it could from, Plato's Heaven? Then to take it a step further, since the axiomatic systems are the form in which scientific knowledge of mathematics is held (formally or informally), the material within them can then be used to form new ideas and concepts leading to new, more abstract systems, without going back explicitly to the starting point in reality with numbers, etc. But implicitly or explicitly, all valid mathematical concepts and principles are rooted in our perception of physical reality.

Tom Rexton: So mathematical concepts, definitions, conjectures, generalizations, axioms, etc. are formed inductively, then they are reorganized, systematized and proved deductively. Still, many mathematicians and philosophers alike tend to belittle the former (induction) and think that mathematics is all about the latter (deduction). They present mathematics as though it all starts with axioms and rules of deductive inference. I suppose this is due a lack of an explicit understanding of the nature of induction and its contribution to science, especially to mathematics.

In particular it has been due to a lack of a proper theory of concepts (starting with the problem of universals), which are of course inductive themselves. You frequently see the presentation of mathematical axioms as based on what are called "intuitive" "motivations", suggested as suspiciously tainted and not true knowledge, only hinting at the axioms (you can see the Platonic influence in this). The "intuitions" are then swepts aside as irrelevent to "actual" mathematics (as in Russell's epistemology of floating hierarchies). Then everyone wonders what the objective basis and actual meaning of mathematics is! Stolen concepts is the least of it -- the whole science is stolen.

It is true that only axioms, theorems and definitions within a system may be used for the deductive development, but it is not true that the original concepts and their meanings are irrelevent to the science as such -- they provide the objective foundation, rooting the whole science in reality as a science of method built on our basic concepts of numbers and spacial relations as means of measurement.

If this thread continues to the end of Landau to real and complex numbers, we will see in detail how a proper conceptual basis answers the traditional "mysteries" of their cognitive status and meaning -- both in the "ordinary" pre-axiomatic realm and in the technical mathematical realm -- and how the traditional understanding and presentation of the postulational approach to the foundations of analysis as floating hierarchies creates the confusion.

[Fesapo]

It is an implicit function definition in that it specifies the function through a general relation.  That is why it must by proved that this relation is in fact a function for all x and y, i.e., that the function exists and is single-valued, with a unique value for each x and y.  (We discussed the uniqueness portion of this above in #66.) 
 
You can't, at this stage of the development, assume that using the inductive principle in a recursive definition is a valid approach; You have to prove it.  The axiom of induction is a rule of proof, which is not the same as definition, even though the axiom of induction and recursive definitions both use the same basic idea.  You have to prove, using the axioms, that using the idea of repeated succession implicitly specified by a recursion does in fact define a function.  Of course it all works out because recursion and the axiom of induction are so similar. 
 
A definition invoking "y number of successions to get to y" is not valid within the axiomatic framework at that stage of its development.  You don't yet have the concept of indexing a sequence, or the concept of order to refer, in general, to all numbers up to a certain number used to reach that number.  See the related discussions in Landau pp. ix-x in the preface, and in Thurston pp. 10-11. 
 
You are giving a pre-axiomatic definition of addition based on counting.  That knowledge is implicit in the formulation and eventually comes out of it, is required to get to the stage of the axiomatic formulation at all, and does not mysteriously disappear from knowledge once you begin analyzing the consequences of Peano's axioms.  But you will not be able to proceed in the development without understanding the abstraction of the recursive definition, which is used in the inductive formulations and proofs in the subsequent development.  Attempting to avoid the recursive definition is like counting on your fingers to avoid the more advanced abstract concepts and methods of addition and subtraction, or the temptation we all had at the beginning to short circuit Euclid by using principles that are "obvious" about triangles instead of following the progression of definitions and proofs using Euclid's axioms and theorems. 
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Happy New Year! I'm ready to start back in on Landau, if you are. I'm still stuck on why it is valid to define addition using an implicit formula. Shouldn't they go ahead and do the development in order to be able to give an explicit definition as I described it?

[ewv]

Fesapo: I'm still stuck on why it is valid to define addition using an implicit formula. Shouldn't they go ahead and do the development in order to be able to give an explicit definition as I described it?

You should forget about "implicit formulas" since that is a wider concept than you need for this. The important concept is that of a "recursive" or "inductive" definition, which is required to apply the principle of mathematical induction in proofs of theorems.

Your attempt to define "x+y as x-with-y-number-of-successions" is also "inductive" in the general philosophical sense of the term. The recursive definition in Landau makes it more explicit in a form that can be used to proceed with the axiomatic development. If you repeatedly apply the recursive formula you get the formula you are thinking of, but you need the more explicit formulation in terms of a specific process of how to go one step at a time in order to apply the rule of mathematical induction in proofs.

As for why it is a valid definition, you have to prove that a formula does in fact define a function by showing that its domain is the entire set and that it is single-valued. That is why Landau calls it both a definition and a theorem -- he proves that it is in fact a valid definition.

You have to be willing to follow the logical train of thought as it is presented. Once you see how it all works and fits together then you can go back and see what happens if you try to omit something, how far you might be able to get with alternative approaches, and what works and what doesn't. (There are differences between Thurston and Landau in the logical progression.) But you can't insist that a science that you do not yet know should have been done differently.

Mathematical induction is used extensively to prove the theorems on addition and multiplication of the natural numbers, and then only rarely, if at all, for the rest of the development in Landau through the real numbers. So you have to get at least through the theorems on addition and multiplication to see the whole story of how the concept of recursive definition is used. Remember that this is only the beginning and the purpose is to get a science of real and complex numbers, which is how you got started on this.

In the meantime, to get a better idea of how the inductive process works for definitions, you should jump ahead and consider the inductive definition of exponentiation (see Thurston p.90, Landau doesn't include it). Exponentiation relies on multiplication, and therefore on addition, so it is farther along in the logical progression of conceptual dependencies, but it is simpler to see how the pattern works for exponents. The exponential function x^y -- x multiplied by itself y times -- is defined inductively as
x^1 = x
x^y' = x*x^y
That specifies explicitly how the process of repeated multiplication works one step at a time. The inductive definitions of addition based on counting and of multiplication as repeated addition do the same thing.

[Fesapo]

You have to be willing to follow the logical train of thought as it is presented.  Once you see how it all works and fits together then you can go back and see what happens if you try to omit something, how far you might be able to get with alternative approaches, and what works and what doesn't.  (There are differences between Thurston and Landau in the logical progression.)  But you can't insist that a science that you do not yet know should have been done differently.

<snip>

In the meantime, to get a better idea of how the inductive process works for definitions, you should jump ahead and consider the inductive definition of exponentiation (see Thurston p.90, Landau doesn't include it).  Exponentiation relies on multiplication, and therefore on addition, so it is farther along in the logical progression of conceptual dependencies, but it is simpler to see how the pattern works for exponents.  The exponential function x^y -- x multiplied by itself y times -- is defined inductively as
x^1 = x
x^y' = x*x^y
That specifies explicitly how the process of repeated multiplication works one step at a time.  The inductive definitions of addition based on counting and of multiplication as repeated addition do the same thing.
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So you are suggesting that I not stick so rigorously to understanding each step in the causal chain, as it builds up? I guess that what bothers me about doing that is just that I want to really understand this and the feedback I'm getting indicates that I'm missing something right at this point. But you're the teacher. If you think that I need to jump ahead, I'll take a look at it.

[jcbaduk]

Hello everyone, I'm new to this forum. I hope I can learn a lot from discussing ideas with serious people. Especially ideas in math and physics, I am so glad that there are Objectivists interested in this area (sorry, but the guys in my campus club only talk about art and history, and I rarely participate because the meeting time conflicts with my classes).

I have been looking through this post, and a question came up to me. In light of cantor's definition of countable and uncountable sets, Objectivists Epistemology book made it clear that zero and infinity are only concepts of method. The question is : If I randomly pick a point between 0 and 1 on the number line, the point must always be a terminating decimal (i.e. no unending decimals, except for 0.333333... and others that are rational numbers) or even irrational numbers (i.e. (root 2)/2 that can be found by bisecting the hypotenuse of a right angle 1 x 1 triangle) or any that can be constructed physically with a compass and a ruler. Thus, all numbers picked on the number line must be countable?

Of course I twisted the definition of "countable" (i.e. by accepting irrational numbers). Basically, as long as it can be physically constructed using a compass and ruler, it should be "counted". What do you think of this definition? However, numbers like pi/3 will never be located on the line, or would it?

I have some doubts as to whether "point", "random" are valid concepts.

[jcbaduk]

Sorry, let me clarify. The reason why the points you choose on the line must be terminating is because there is no such thing as infinite decimals. Irrational numbers like (root 2)/2 are exceptions because they can be physically constructed. Also, in light of the article "The unbounded, finite universe", why can't we similarly say that these "infinite decimals" are finite numbers with unbounded decimals?

[ewv]

Hello everyone, I'm new to this forum. I hope I can learn a lot from discussing ideas with serious people. Especially ideas in math and physics, I am so glad that there are Objectivists interested in this area (sorry, but the guys in my campus club only talk about art and history, and I rarely participate because the meeting time conflicts with my classes). I have been looking through this post, and a question came up to me.

In light of cantor's definition of countable and uncountable sets, Objectivists Epistemology book...

Welcome to the Forum jcbaduk.

I'm assuming that the book you mean is her Introduction to Objectivist Epistemology.

...made it clear that zero and infinity are only concepts of method.

Yes, all concepts of mathematics are concepts of method. But that is completely independent of Cantor's theories, not "in light" of the them.

The question is : If I randomly pick a point between 0 and 1 on the number line, the point must always be a terminating decimal (i.e. no unending decimals, except for 0.333333... and others that are rational numbers) or even irrational numbers (i.e. (root 2)/2 that can be found by bisecting the hypotenuse of a right angle 1 x 1 triangle) or any that can be constructed physically with a compass and a ruler.

It depends on how you "pick" it. If you do it by some physical measurement you can only measure to some finite precision. If you "pick" it by some algorithm -- including a geometric construction method referring in principle to some number -- it can refer to a number with an infinite sequence of unending decimals such sqrt(2), but only as an abstraction.

Thus, all numbers picked on the number line must be countable?

"Countable" technically refers to the enumerability of the set, not a particular number or its precision. See this thread here on the forum.

Of course I twisted the definition of "countable" (i.e. by accepting irrational numbers). Basically, as long as it can be physically constructed using a compass and ruler, it should be "counted". What do you think of this definition? However, numbers like pi/3 will never be located on the line, or would it?

"Countable" has a technical meaning (see the link above). You can "construct" irrational numbers with a compass and ruler, but you can't do it to infinite precision. The abstract methods you use to specify numbers by potential "constructions" do not distinguish between countable and uncountable.

I have some doubts as to whether "point", "random" are valid concepts.

They are valid concepts as abstractions. A point does not mean something exists without extent. You abstract away its size as negligible. "Random" has an abstract technical meaning; it does not mean "causeless".

Sorry, let me clarify. The reason why the points you choose on the line must be terminating is because there is no such thing as infinite decimals. Irrational numbers like (root 2)/2 are exceptions because they can be physically constructed.

You can physically "construct" sqrt(2) to any precision in principle; you cannot construct or measure anything to infinite precision. When you "construct" sqrt(2) you have established in principle what is in fact an unending decimal, but any particular instance of a measurement will have a finite precision and therefore a finite number of decimal places. Sqrt(2) is not an exception. It is a number which when squared is two within some precision, what the particular precision being left unspecified. When you deal with it in some specific context, the relevant precision is determined.

Also, in light of the article "The unbounded, finite universe", why can't we similarly say that these "infinite decimals" are finite numbers with unbounded decimals?

I don't know what that means. Every number is a finite number. The extent of its decimal expansion is its precision. Numbers with unbounded decimal expansions are abstractions, with their precision left unspecified.

[jcbaduk]

Thank you evw for the reply. It makes things a lot clearer. From what I understand, it seems that there's no such thing as infinite precision. Thus, while it would be possible to construct a triangle with 1, 1, root 2 as its side as an abstraction, it is not possible to construct them in practice. However, doesn't that imply that what's possible in theory (i.e. abstraction), is not possible in practice (i.e construction)? Is this a false dichotomy, or am I using a stolen concept?

The "unbounded finite universe " article says that it's not necessary to talk about the "size" of the universe since universe came before size. Since existence exist finitely, the universe is finite. It goes to show that since we don't have to talk about its size, the universe does not need to be bounded while being finite. Similarly, on the concept of precision for numbers, parallel to the concept of "size", since precision comes after numbers (is this wrong?), Would it be wrong to ask for precision when "constructing" numbers like root 2? Shouldn't we accept the decimals of root 2 as "unbounded"? What makes it impossible to construct numbers like root 2? We don't have to measure root 2, we just simply construct it by drawing that triangle. This is an analogy I thought of immediately. However, it still bothers me to see why it's okay to lose "precision" in abstractions like root 2 but not when you construct them.
Perhaps precision is only a valid concept in measurement?


Thanks for telling me about the thread on countable sets. I'm still reading it. I would really like to find an algorithm to enumerate all lengths constructible using a compass and ruler using the discrete units of delta, where delta approaches the limit of 0.

[Paul's Here]

There seems to be an assumption being made about the physical representation of a 1, 1, root 2 triangle. Just as the root 2 hypotenuse can only be measured to a specific precision and cannot have an actual infinite precision, so the 1 sides can only be measured to a specific precision and cannot have infinite precision. The 1 side is not accurate to infinite 0's: 1.0000---. It is only as accurate as the instrument you use to make a measurement. So the issue of a phyiscal measurement actually being root 2 makes no more sense than a physical measurement actually being 1.000---.

[ewv]

There is no such thing as an infinitely thin line, a line that is straight to infinite precision, or infinitely sharp points at the ends of a line or vertices of a triangle. These are all abstractions in which certain attributes are ignored as negligible. The whole 'construction' of a 'perfect' right triangle is an abstraction. If one of the legs of the triangle is the standard of a unit of measurement (with length 1), it can only be measured or duplicated as a second leg of length one, and at a right angle to the first, to finite precision. So nothing has been "lost" in the finite precision of the length of the hypotenuse as such when you construct or measure it.

[ewv]

Thank you evw for the reply. It makes things a lot clearer. From what I understand, it seems that there's no such thing as infinite precision.

There is no such thing as an "infinite" anything other than as an abstraction referring to a mathematical potential. An infinite precision would require an infinite subdivision of the base unit, which is impossible. That doesn't mean that there is no such thing as 'absolute precision' or an 'exact measurement'. See "Exact Measurement and Continuity" in Introduction to Objectivist Epistemology, Appendix -- Measurement, Unit, and Mathematics.

Thus, while it would be possible to construct a triangle with 1, 1, root 2 as its side as an abstraction, it is not possible to construct them in practice. However, doesn't that imply that what's possible in theory (i.e. abstraction), is not possible in practice (i.e construction)? Is this a false dichotomy, or am I using a stolen concept?

You don't "in practice" construct the triangle "as an abstraction"; you construct an actual triangle to some degree of precision. The theory which says you can do that is correct. 'In principle', you can do it to any degree of precision, but it must be to some definite degree of precision. You can't do to it to any degree of precision with an actual macrosopic ruler and compass made out of wood and metal, etc.: the 'construction' process in general is itself a mathematical abstraction of intersecting circles with lines, etc. But that subsumes all you know about the meaning of points, lines, precision, infinity, decimal expansions, etc. (Though historically the Greeks did it only with the geometric concepts, not decimals.) That knowledge includes what you mean by abstracting away the extent of a point and the thickness of a line as negligible, but not zero, etc.

When you properly use the abstract concepts of points, lines, etc. without explicit mention of the omitted measurements, you are operating on a higher level of abstraction, using abstract concepts as the units of your thought. That doesn't mean that all of the things you had to know to get to that level of thought no longer exist. They are still subsumed in the meaning of your concepts, even though you don't dwell on them consciously when that is not required. The "theory" does not mean that somehow out in reality there is supposed to be a line of length sqrt(2) with infinite decimal places and with complete disregard for the abstract nature of points and lines. The theory is a mathematical method conceptually relying on a limiting process as points, lines, etc. become 'ideal'.

So if by "construct as an abstraction" you mean that abstract mathematical process, it doesn't mean you can do something "in theory" but "not in practice" as some kind of dichotomy divorcing theory from practice. But if you misinterpret the theory to reify the abstractions in a Platonic mathematics, thinking that there are points with no extent, lines with no thickness, lines that are straight to infinite precision, etc., then you are in trouble. But that isn't because of the incommenurablility of the sqrt(2); with a Platonist approach to abstractions you're in trouble long before that!

The "unbounded finite universe " article says that it's not necessary to talk about the "size" of the universe since universe came before size. Since existence exist finitely, the universe is finite. It goes to show that since we don't have to talk about its size, the universe does not need to be bounded while being finite.

Be careful with metaphysical speculation about the physical universe beyond what we know, and keep it separate from the epistemology of the mathematical concept of infinity as a concept of method.

Similarly, on the concept of precision for numbers, parallel to the concept of "size", since precision comes after numbers (is this wrong?),...

The concept of numerical precision comes after the concept of number. You don't get to it until you start dealing with subdivisions of units, i.e., fractions.

... Would it be wrong to ask for precision when "constructing" numbers like root 2?

No. Even with a ruler and compass, if you use smaller points and thinner lines and are more careful you get more precision.

Shouldn't we accept the decimals of root 2 as "unbounded"?

Not if you mean it metaphysically. The sqrt(2) is an abstraction; it does not exist prior to concepts of number and measurement. There is no infinite subdivision of units out there waiting to be identified. That is the reverse of the actual process. There is a length there and you subdivide to measure it with increasing precision.

Mathematically, the number of decimal places increases without bound as the precision increases, but that is an abstract mathematical process of method. The quantity sqrt(2) is bounded no matter how many decimal places you use because the sequence has a mathematical limit.

What makes it impossible to construct numbers like root 2?

Nothing, it's a number which when squared is 2 within some precision. You can't get around the precision with a physical construction or measurement because everything you do has some finite precision; the lead in your compass has some thickness, etc.

But sqrt(2) also has a technical meaning of a unique abstraction referring to a limit in the real number system. Fully understanding that abstraction, which is more technical, requires understanding both limit processes and the axiomatic mathematical construction of the real number system in which numbers like sqrt(2) which do not exist the way rational numbers do are constructed as a higher level abstraction and regarded as if they were numbers at that level.

You cannot 'construct' and abstraction like the sqrt(2) because you can't reify an abstraction. The mathematical 'construction' using a triangle with abstract points and circles only has meaning as an abstraction; there is no Platonic sqrt(2) to construct or find out there someplace in a Platonic triangle.

We don't have to measure root 2, we just simply construct it by drawing that triangle. This is an analogy I thought of immediately. However, it still bothers me to see why it's okay to lose "precision" in abstractions like root 2 but not when you construct them.

You don't lose it. There is nothing to lose. You start out and live in a world in which everything is what it is, is something in particular, and is therefore definite and specific, i.e., finite, in every respect. All real triangles have lines of some thickness, splotchy vertices and sides that aren't infinitely straight. There are no infinities to lose when you draw a triangle in search of an 'ideal' sqrt(2) because they don't exist.

The Pythagorean theorem is a mathematical principle of method which applies to numerical measurements which by their nature are finite in every way; it is not a metaphysical principle predicting infinite subdivisions in reality. If you get the hierarchy of concepts right and the concepts properly defined and understood, including the concepts required before the formal mathematics begins, you will see that the search for a 'construction' of a length corresponding to sqrt(2) with infinite decimal places is the search for a stolen concept by means of floating abstractions. You also see the facts and the conceptual hierarchy that give rise to the necessity of the abstract mathematical concepts, including infinity.

Perhaps precision is only a valid concept in measurement?

No. The hypotenuse you are looking to construct is what it is, but it is part of a finite triangle relating finite parts. Precision is as much a part of its active construction as its measurement.

Thanks for telling me about the thread on countable sets. I'm still reading it. I would really like to find an algorithm to enumerate all lengths constructible using a compass and ruler using the discrete units of delta, where delta approaches the limit of 0.

Constructability usually technically means with a finite number of steps, not an unlimited sequence of operations. If you allow an open-ended procedure you are duplicating the limit process and can 'construct' any real number by adding on sucessive subdivisions.

[inventor]

There seems to be an assumption being made about the physical representation of a 1, 1, root 2 triangle. Just as the root 2 hypotenuse can only be measured to a specific precision and cannot have an actual infinite precision, so the 1 sides can only be measured to a specific precision and cannot have infinite precision. The 1 side is not accurate to infinite 0's: 1.0000---. It is only as accurate as the instrument you use to make a measurement. So the issue of a phyiscal measurement actually being root 2 makes no more sense than a physical measurement actually being 1.000---.

Nothing cannot have infinite precision.

Inventor