martjoh

The so-called "infinity" in mathematics

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It is a fact that infinity is an invalid concept. But what then is the so-called "infinity" in mathematics?

I have an idea on my own on what it really is: Precision omission. I can elaborate further on that if there is any interrest. But first, I would like to get your thoughs on the question of why it works.

A good example of using "infinity" is the area of a circle. It is evaluated to pi*r*r (r=radius) by fitting a triangle inside a circle, then a square, then a pentagon, then a 6-gon, all the way to an "infinity"-gon, at which point you have the correct area... By precision omission I mean the "infinity" in the "infinity"-gon. e.g. (example-values are taken at random) 5-gon will give you the area with 0.1 error margin, and a 6-gon 0.01 etc. Hence, the actual substitute for "infinity" for a precision of 5 is 0.1 and for 6, 0.01 etc.

- martjoh

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It is a fact that infinity is an invalid concept.

Yes, as a metaphysical existent, without a doubt.

But what then is the so-called "infinity" in mathematics?

The usual answer is that it is a concept of method, representing a potential, not an actual. (However, as an aside, Harry Binswanger has argued against accepting infinity even as a valid concept of method.)

I have an idea on my own on what it really is: Precision omission. I can elaborate further on that if there is any interrest. But first, I would like to get your thoughs on the question of why it works.

A good example of using "infinity" is the area of a circle. It is evaluated to pi*r*r (r=radius) by fitting a triangle inside a circle, then a square, then a pentagon, then a 6-gon, all the way to an "infinity"-gon, at which point you have the correct area... By precision omission I mean the "infinity" in the "infinity"-gon. e.g. (example-values are taken at random) 5-gon will give you the area with 0.1 error margin, and a 6-gon 0.01 etc. Hence, the actual substitute for "infinity" for a precision of 5 is 0.1 and for 6, 0.01 etc.

Perhaps I am missing your point, but this is just a process converging to a limit, a process which you stop at intermediate points. Rather than being "omitted," precision is being established.

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Perhaps I am missing your point, but this is just a process converging to a limit, a process which you stop at intermediate points. Rather than being "omitted," precision is being established.

I based the conclusion on the fact that every result in mathematics, every application of it, must have a certain precision, but can have any precision. The only difference between 2.0000... and 3.1415.... (pi) is the method by which the digits are calculated. But there must be some amount of digits, i.e. some precision.

There is a difference between the measurement 2 meter and (e.g) the factor 2 in a forumla. The first is a measurement which requires a precision, the latter is a method, a doubling, which itself does not require a precision.

Common to all occurances of "infinity" are two things: (1) The higher number you substitute for "infinity" the higher the precision. (2) For every measurement, i.e. application of a formula, a number is substituted for "infinity" in order to make the calculation. i.e. You use some amount of digits of pi when you calculate the area of a given circle, you cannot use all of them.

Hence, given a final measurement, all numbers used in the calculation are finite, whole numbers (decimals are rational numbers a/:excl:. And where "infinity" was found in the formulae, a number is substituted to provide the required precision, i.e. to gain 7 digits of pi, which was omitted in the formula.

- martjoh

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I based the conclusion on ...

Martin, all measurement implies some degree of precision, so I just do not see any important significance linking this with infinity. If infinity is accepted (not all do) as a proper concept of method, then its essential element is the potentiality of a process, whatever mathematical process that may be. Infinity is not some very big number, it is a process, and I fail to see any connection there to "precision omission."

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Hence, given a final measurement, all numbers used in the calculation are finite, whole numbers (decimals are rational numbers a/:excl:. (martjoh)

Not all decimal numbers are rational eg. sqrt(2), sqrt(3).

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Martin, all measurement implies some degree of precision, so I just do not see any important significance linking this with infinity. If infinity is accepted (not all do) as a proper concept of method, then its essential element is the potentiality of a process, whatever mathematical process that may be. Infinity is not some very big number, it is a process, and I fail to see any connection there to "precision omission."

Instead of using infinity, I would say the following: For every precision of the measurement of the area of a circle, there is a corresponding n-gon. One would normally say, the circle may be represented by an infinity-gon.

If you where to measure the area of a circle, you would need two things: A radius and a precision. And you are right, the radius always includes a precision, but that precision must be chosen in advance, and then applied to find the number of digits of pi necessary, so that you would achieve the right precision in the final measurement of the area. And since you *have* to fill it inn, I would say it had to be omitted.

Not all decimal numbers are rational eg. sqrt(2), sqrt(3).

markschersch, I think you ment "real numbers" not decimal numbers? You are right, the numbers you gave cannot be represented as rationals, i.e. a/b, but I said decimals, which is always a rational number.

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I would say the following ...

I'm sorry, Martin, but I just cannot follow your point, so it is best if I just leave it be.

... but I said decimals, which is always a rational number.

Non-terminating, non-repeating decimals are irrational numbers.

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decimals are rational numbers a/b
As I understand the terminology, "decimal (number)" refers to a base-10 number (as opposed to binary, octal etc). A "decimal fraction", OTOH, is a fraction with a power of 10 as the denominator.

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I'm sorry, Martin, but I just cannot follow your point, so it is best if I just leave it be.

Ok, thanks anyway. Its great to be able to discuss this. A common professor would shrugg it off and mumble something about 1 to 1 correspondance of sets and aleph numbers. At least we're on the same base.

As I understand the terminology, "decimal (number)" refers to a base-10 number (as opposed to binary, octal etc). A "decimal fraction", OTOH, is a fraction with a power of 10 as the denominator.

I was a bit unclear here. But I meant numbers like 1.243 (decimal) may always be written as 1243/1000 (rational). The base is unimportant really (wether binary, octal or decimal)

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Ok, thanks anyway. Its great to be able to discuss this.

Since I love the entire subject, I feel that way too. It is frustrating, however, not to be able to wrap my mind around a thought or an idea. Perhaps if you think about it some more you may be able to express your idea differently, and then maybe your view will be better received.

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From what I understand, the mathematical concept of infinity is as was explained in post #2, a "concept of method". Infinity is not anyTHING, it is the process of increasing THINGS (numbers, in the case of math) without termination. Though it can be treated in some respects as if it were a number, one should always keep full context and remember infinity as a process. This may be similar to the use of the imaginary operator and other complex numbers; they can be used as conceptual tools but the user must always remember they are not real things (though infinity is not real for different reasons than complex numbers; infinity is not real because it is not a thing, and I don't even know if it is a REAL process, or what that would even mean, while complex numbers are imaginary things).

I think many students do not grasp the complexities of dealing with infinity, and while I do not claim to understand it fully either, I recognize that there are complexities to be understood.

One of my intended majors is mathematics, and as such I'd appreciate any recommended books/articles on the subject; Stephen Speicher, I'm particularly interested in Binswanger's argument, have you a link to an article?

In general I think Wikipedia and MathWorld can be used as good "general" sources on many subjects in math -- a lot of it is pretty neat too, if I may shamelessly plug my second favorite subject.

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One of my intended majors is mathematics, and as such I'd appreciate any recommended books/articles on the subject; Stephen Speicher, I'm particularly interested in Binswanger's argument, have you a link to an article?

Hmm. One of your majors? You sound ambitious, which I like. :excl: What are your other(s)?

As to books, there are none that will provide the proper perspective on this issue. However, a great deal of fascinating work on this exists, in historical papers and texts, right up to modern day work. One such interesting work was that done by Abraham Robinson, who some forty years ago developed what is now known as nonstandard analysis, in which he established a rigorous foundation to the vague notion of infinitesimals devised by Newton centuries ago. I would not advise reading his work directly until you get further into your mathematical studies, but you can get a really nice overview of the man and his accomplishments in an excellent biography of him, Abraham Robinson: The Creation of Nonstandard Analysis, Joseph Warren Dauben, Princeton Umiversity Press, 1995.

As to Dr. Binswanger's argument: He hinted at it in his Selected Topics in the Philosophy of Science, and he explicated it further in a number of discussions we had on his HBL forum a number of years ago. In a nutshell he argues that the infinite is an invalid concept, in that counting numbers lose their meaning beyond a certain range; a range beyond which we cannot physically represent a number, and it therefore becomes meaningless, and a range by which our own epistemologies can sensbily grasp, by whatever notation, the limited, as opposed to the unlimited. In addition, there is also a physically practical limit beyond which we cannot even represent numbers in any notation. There will be some point which is reached where there are not even enough particles in the universe where the most compact notation could denote and delimit a number. (Any errors in this representation are my own.) The conclusion, then, is that the number system is not even potentially infinite.

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From what I understand, the mathematical concept of infinity is as was explained in post #2, a "concept of method". Infinity is not anyTHING, it is the process of increasing THINGS (numbers, in the case of math) without termination. Though it can be treated in some respects as if it were a number, one should always keep full context and remember infinity as a process. This may be similar to the use of the imaginary operator and other complex numbers; they can be used as conceptual tools but the user must always remember they are not real things (though infinity is not real for different reasons than complex numbers; infinity is not real because it is not a thing, and I don't even know if it is a REAL process, or what that would even mean, while complex numbers are imaginary things).

Why complex numbers? Their origin, the negative numbers, are not any more natural. (in physics the negative sign often denotes direction). As AR said in IOE (in my words): Given the basic axiom of math, the arithmetic series, and proper logic, what you derive is valid.

As to Dr. Binswanger's argument: He hinted at it in his Selected Topics in the Philosophy of Science, and he explicated it further in a number of discussions we had on his HBL forum a number of years ago. In a nutshell he argues that the infinite is an invalid concept, in that counting numbers lose their meaning beyond a certain range; a range beyond which we cannot physically represent a number, and it therefore becomes meaningless, and a range by which our own epistemologies can sensbily grasp, by whatever notation, the limited, as opposed to the unlimited. In addition, there is also a physically practical limit beyond which we cannot even represent numbers in any notation. There will be some point which is reached where there are not even enough particles in the universe where the most compact notation could denote and delimit a number. (Any errors in this representation are my own.) The conclusion, then, is that the number system is not even potentially infinite.

It is possible to write out never ending numbers as a formula, if you plot 1/x on a calculator, it goes up close to zero, how far? Hence we have a construct which symbolizes infinity "1/0". Very similar to the construct (-1)^(1/2) (squareroot of minus 1). The former is not "undefined", as every construct in mathematics must have some meaning. (the alledged proofs of "1/0" being undifined, proves the invalidity something else, logically)

In answering Zeno's paradox, infinity is taken as the solution to the following equation: n = n + 1, which is easy to shrugg off at first glance, but it is used dayly in answering the paradox! The "1" might me any non-zero number. I can write out the disproof of Zeno's paradox in my next post to get straight to the point of applying it.

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Why complex numbers? Their origin, the negative numbers, are not any more natural. (in physics the negative sign often denotes direction).

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What do you mean by "natural" in this context? Are negative numbers less natural because they do not represent a quantity of something you can hold in your hand (that is, a positive quantity)? Combined with the generalization of unsigned "normal numbers" to positive numbers (in which no leading sign implies a positive sign), signed numbers represent a quantity relative to some other reference quantity, and are therefore more advanced than "natural" numbers or unsigned rational numbers, but I do not see how that makes them less natural.

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Why complex numbers? Their origin, the negative numbers, are not any more natural. (in physics the negative sign often denotes direction).

When you look for origination look for the essentials. The origin of complex numbers lies in a mathematical process, not the incidental placement of a negative sign. Leaving aside the early Ancient bumblings, in 1545 Gerolamo Cardano in his Ars Magna solved the problem of finding which two parts of the number 10 result in a product of 40. The equation to solve is x(10-x) = 40, which roots he found to be 5 +/- sqrt(-15), which product he then multiplied to verify as 40. This is what led to third and fourth degree equations and eventually to what is now known as Cardan's method.

As AR said in IOE (in my words): Given the basic axiom of math, the arithmetic series, and proper logic, what you derive is valid.

That was not quite her point. Better to let Ayn Rand speak for herself.

It is possible to write out never ending numbers as a formula, if you plot 1/x on a calculator, it goes up close to zero, how far? Hence we have a construct which symbolizes infinity "1/0".

And how is this relevant to the argument I presented?

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Hmm. One of your majors? You sound ambitious, which I like.  :excl:  What are your other(s)?

I'm certain on math, and I am undecided on another major/minor. At different times I've considered philosophy, computer science, and some work in psychology; ideally I'd like to work in the field of "cognitive science" (scare quotes because everything that currently exists in the "field", save some of Searle's work, is complete trash) and initiate or work on projects that actually do something to help understand the human mind (or animal mind). I don't have to decide until next fall or soon after, when I enroll (it also depends on where I go, it is difficult to work multiple academic foci into some college's curricula), but it's never to early to think about it.

As to Dr. Binswanger's argument...

That is an interesting argument, but not one I hadn't considered (unfortunately I don't think I know how to completely resolve it). The argument of running out of particles to denote numbers presumes that a certain quantity of particles exist; I don't want to turn this into the eternal/e-length-al (I think that Alex used some similar word, possibly e-size-al in his great essay; quite funny and descriptive) universe argument, though the two are quite intimately related (that is, infinity and the universe). I'm not about to say the universe has an infinite amount of particles (which wouldn't even make sense), but I'm not prepared to say it has "x" amount, either. First I would have to resolve this. So far, I've thought of this thought experiment: imagine a sphere increasing in size from one point in space. In my opinion, the sphere would continue to increase in size without ever not being able to increase in size or ever touching particles it had already touched; that is, at no point in time would it not have new particles to touch. Maybe the word "limitless" is appropriate, but I'm not even sure if that is valid... so obviously I'd have to sort this out.

One more thing, concerning the concept of infinity. In concept creation, we omit measurements, so is there any way we can retain infinity as a valid concept, even if just as a process, by omitting the measurement of the supposedly exhaustable number of universal particles? That is, infinity can still be thought of as an increase in something (an increase in nothing in particular would be invalid for sure), but something that can increase without any particular restraint? I've always preferred the mathematical nomenclature of "arbitrarily large" that my teachers/professors use, and it could be hinting at a possibly valid construction of the concept.

Besides in higher level math, infinity and the subsequent debates about it remain minutiae, but I still think it is an appropriate topic for discussion, if for no other purpose than periodic mental exercise and entertainment.

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I'm certain on math, and I am undecided on another major/minor.  At different times I've considered philosophy, computer science, and some work in psychology; ideally I'd like to work in the field of "cognitive science" (scare quotes because everything that currently exists in the "field", save some of Searle's work, is complete trash) and initiate or work on projects that actually do something to help understand the human mind (or animal mind).

I would agree that, to the degree that an area in the "cogntive sciences" relies upon an understanding of the working of consciousness, then to that degree the field is greatly bankrupt. On the other hand, to the degree that an area of the "cognitive sciences" acts more as a hard science, such as can be seen in many aspects of neuroscience, then to that degree there are remarkable things being done.

Personally, rather than discourage, I would encourage bright young minds with an Objectivist perspective on consciousness to enter the cognitive sciences, on whatever aspect or level they find most intriguing. The way to wash away the bad parts of a field, is to flood it with good parts. There are endless fascinating challenges awaiting good people in this field.

That is an interesting argument, but not one I hadn't considered (unfortunately I don't think I know how to completely resolve it).  The argument of running out of particles to denote numbers presumes that a certain quantity of particles exist; I don't want to turn this into the eternal/e-length-al (I think that Alex used some similar word, possibly e-size-al in his great essay; quite funny and descriptive) universe argument, though the two are quite intimately related (that is, infinity and the universe).    I'm not about to say the universe has an infinite amount of particles (which wouldn't even make sense), but I'm not prepared to say it has "x" amount, either.  First I would have to resolve this.

(Alex uses "asizal.") That is an interesting point you make, and I will have to think about it, but the physical part of the argument was not as important as the epistemological part, or so I think.

So far, I've thought of this thought experiment:  imagine a sphere increasing in size from one point in space.  In my opinion, the sphere would continue to increase in size without ever not being able to increase in size or ever touching particles it had already touched; that is, at no point in time would it not have new particles to touch.  Maybe the word "limitless" is appropriate, but I'm not even sure if that is valid... so obviously I'd have to sort this out.

If you mean this as a physical analogy to the universe, then I do not see how this applies. You cannot physicalize the issue.

One more thing, concerning the concept of infinity.  In concept creation, we omit measurements, so is there any way we can retain infinity as a valid concept, even if just as a process, by omitting the measurement of the supposedly exhaustable number of universal particles?  That is, infinity can still be thought of as an increase in something (an increase in nothing in particular would be invalid for sure), but something that can increase without any particular restraint?

If you are here solely referring to infinity as a concept of method, then the measurements omitted would relate to the particular parts or steps of whatever the process you define. The argument from exhaustion of actual particles was part of an attempt to show that even in principle the method could not work, thereby obviating the "potential" as well as the actual part of infinity.

I've always preferred the mathematical nomenclature of "arbitrarily large" that my teachers/professors use, and it could be hinting at a possibly valid construction of the concept.

I previously mentioned Abraham Robinson's work in this regard, and there are others. The really hard part is establishing a foundational basis for it all, starting with the primitives.

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I'm replying to the thread as a whole. In particular the notion of infinity as not a valid concept of method.

I understand Dr. Binswanger's argument that no matter what symbolism is employed you will at some point reach a limit in the ability to represent a (large enough) quantity. I believe that this argument does not depend on the universe consisting of an unlimited number of particles. I think it can be made just by the fact that our minds are limited.

Proceeding from this I am left wondering about the use of diagonalization as a inductive proof technique.

The classic use of diagonalization is to prove that there are more real numbers than natural numbers. My question is, if we banish infinity as a concept of method then is diagonalization still a valid inductive method?

My answer is that the diagonalization method is valid, but does not show that one infinite set is larger than another infinite set, but rather shows that one set is countable verus not countable. Countable means an algorithm can be specified to enumerate over the set. Integers and natural numbers have this property, but real numbers do not.

I think a big problem with modern mathematics today is the reification of "mental sets of objects". As a result of this sloppy thinking, infinity gets used when another terminology can better be used to express the idea.

PROOF THAT THERE ARE MORE REALS THAN INTEGERS:

(See Roger Penrose, The Emperor's New Mind)

Assume that every real number can be counted by an integer. Then every real number between 0 and 1 can also be mapped to an integer. Such a mapping would look something like this:

1 = 0.12345 ...

2 = 0.89567 ...

3 = 0.45256 ...

4 = 0.72515 ...

5 = 0.80204 ...

etc...

Consider the diagonal digits from the list:

1, 9, 2, 1, 4, ...

The diagonal digits from our list can be used to form various wierd and wonderful real numbers. The first obvious one is this:

0.19214 ...

But a contradiction arises if we define a real from our list such as:

0.21121 ...

How was this real number generated from the diagonal digits? Using the following rule: if the diagonal digit is '1' then place a '2' in the corresponding spot in the derived real number. If the diagonal digit not '1' (2 through 9 and 0), then place a '1' in the corresponding spot. This gives,

0.21121...

Does this number exist in our mapping? No it does not. And cannot exist in the list and yet it is a real number and we assumed all real numbers between 0 and 1 could be expressed by an integer. So our assumption is wrong. Real numbers cannot be counted by integers.

Why is 0.21121... not in the list? Because it cannot match any of the numbers in the list by definition. It has at least one digit different.

No matter what countable procedure you invent to map real numbers to integers, somebody can always invent another procedure that generates a real numeber that won't be in the mapping.

The conclusion is that the set of real numbers is greater than the set of integers. This is an example of sloppy thinking in my opinion. It is worded so that one thinks that there is a set of numbers sitting somewhere that you could look at. The set of integers is really an algorithm for churning out symbols in an ordered way from the previous symbol. Real numbers don't have this kind of countability.

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...infinity can still be thought of as an increase in something (an increase in nothing in particular would be invalid for sure)...

This brings up a point that I think has been hinted at here, but not explicitly expressed. As I understand numerical concepts in Objectivism, a numerical concept refers to either a certain number of existents or a specific measurement. The type(s) of existent or measurement unit is omitted in the formation of the concept, but must be present in reality. So, for example, the concept of three has no meaning, no existance, all by itself. It refers to three apples, three cats, three inches, three degrees Celcius, or three of any other existant or unit of measurement, but it must refer to some existant count or unit of measurement. We often say things like "2+3=5" and omit the referent either for convenience or because we just don't care at the moment (as is often the case when teaching or researching mathematics). But, if we are talking about reality, a unit of some sort must be specified.

The result of this is that for any given problem, experiment, etc. that one is involved with in reality, the maximum value used (or the minimum value on the micro side of things) will be dictated by reality. To me, infinity is invalid as a concept of method, not because we eventually couldn't write the number down, but because given any particular context, there is always a limit in reality. The purpose of a method is to find the answer to some problem in some context. Once the problem and context is named, infinity doesn't apply. For example, one may be working on a problem that requires the calculation or use of the number of Planck distances from one side of the observable universe to the other at a given time; a gargantuan number, but a finite one in the context of one's problem.

Just because we can drop context and imagine one more than a given number doesn't necessarily make the new number "valid" or real. When dealing with reality, it eventually becomes necessary to ask "n=n+1 of what?" and demand an answer, and any valid (i.e., non-imaginary) answer given will dictate a finite range for n.

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The result of this is that for any given problem, experiment, etc. that one is involved with in reality, the maximum value used (or the minimum value on the micro side of things) will be dictated by reality. To me, infinity is invalid as a concept of method, not because we eventually couldn't write the number down, but because given any particular context, there is always a limit in reality.

A "limit in reality" seems to be more an argument against the existence of an actual physical infinity, than against a concept of method. The notion of a concept of method is as a potential process, not an actual one. There is a difference between mentally sub-dividing a line into ever smaller units, as compared to actually performing the sub-division in physical reality.

The purpose of a method is to find the answer to some problem in some context. Once the problem and context is named, infinity doesn't apply.

Well, I will name a simple problem and its context as a counterexample to your claim. (Note that this is just one of an endless stream of such examples in mathematical physics). The absorption of high-energy radiation, such as gamma rays, can cause a nuclear disintegration, or transformation, generally referred to as photodisintegration. A simple nuclear system is a deuteron, a bound system consisting of two nucleons. That is the context. The problem is to calculate the reaction of deuteron photodisintegration. (This is a real, practical problem, one which confirmed the predictions of relativity some 70 years ago.)

In order to perform the calculation, in part we use a perturbation theory technique where a complex function (the Hamiltonian) is integrated between the limits of plus and minus infinity. There is no correspondence between the infinity technique and anything in physical reality, but it is a perfectly valid concept of method which does, in fact, apply to the problem and its context. And, in fact, it does provide a very useful answer.

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After getting some replies, I will try to refrase my take on infinity in terms of the replies:

I would like to continue the reasoning of Jonathan Powers from this paragraph:

This brings up a point that I think has been hinted at here, but not explicitly expressed. As I understand numerical concepts in Objectivism, a numerical concept refers to either a certain number of existents or a specific measurement. The type(s) of existent or measurement unit is omitted in the formation of the concept, but must be present in reality. So, for example, the concept of three has no meaning, no existance, all by itself. It refers to three apples, three cats, three inches, three degrees Celcius, or three of any other existant or unit of measurement, but it must refer to some existant count or unit of measurement. We often say things like "2+3=5" and omit the referent either for convenience or because we just don't care at the moment (as is often the case when teaching or researching mathematics). But, if we are talking about reality, a unit of some sort must be specified.

And then use Stephen Speicher's example

...

In order to perform the calculation, in part we use a perturbation theory technique where a complex function (the Hamiltonian) is integrated between the limits of plus and minus infinity. There is no correspondence between the infinity technique and anything in physical reality, but it is a perfectly valid concept of method which does, in fact, apply to the problem and its context. And, in fact, it does provide a very useful answer.

This line is important here: "Integrated between the limits of plus and minus infinity."

Integration is "infinately" dense summing (right?) to get the area under the graph of the function in question.

If it is possible to calculate the area from -infinity to +infinity, then the function must be decreasing as you move further from 0 (oscillating function decreases periodically etc) (not the only condition, but the essential here) (right?).

When someone uses the formula in Stephen Speicher's example, say to verify it experimentaly, the unit is specified (as Jonathan Powers discussed) and the result naturally has some precision.

If you now trace things backwards, observe the following: The integration is no longer an infinately dense summing, but a summing of some finite density. And the integration is no longer between -infinity and +infinity, but between some finite numbers. (right?)

So, after filling inn all the blanks of the formula, you have a formula of finite, numbers. And where infinity was before, there is now a finite number which will give the required precision. Then, and only then can you evaluate the result.

One question which may arise here is:

But when the formula is used, one does not specify the how dense the summing is, etc, explicitly.

And that is true. But the value is given implicitly when you make a decision on which precision to use. i.e. it is a function of the precision.

Further, it is valid to use "infinity" as we use it today, because we will replace it with a large-enough number whenever we apply it.

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When someone uses the formula in Stephen Speicher's example, say to verify it experimentaly, the unit is specified (as Jonathan Powers discussed) and the result naturally has some precision.

If you now trace things backwards, observe the following: The integration is no longer an infinately dense summing, but a summing of some finite density. And the integration is no longer between -infinity and +infinity, but between some finite numbers. (right?)

I think you missed the point, Martin. The aspect of the formula I referred to does not correspond to anything real in physical reality. That is why we call it a concept of method. Whatever you do in the physical world after the calculation, has nothing whatsoever to do with the calculation itself. Look, I'm sure you must be familiar with what is (mistakenly) called the imaginary number "i." It is used everyday in calculations, by engineers and scientists alike, but you simply cannot point to it in physical reality. It is a concept of method.

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Excuse me, my previous message was messed up by the emoticons.

If it were actually the case that infinite numbers "do not exist", then I would expect the assumption that they do exist which is regularly used by mathematicians to lead to a contradiction. But it has not, as far as I know.

Similarly, if it were actually the case that sufficiently large finite numbers "do not exist", then I would expect ultra-finitism to be the mathematical mainstream rather than a back-water.

As for complex numbers, these are as real as points on a plane. Consider x+iy to be the point (x,y). Then addition of complex numbers is the same as vector addition in the plane. To multiply (a,B) by (c,d), imagine making a copy of the plane and stretching it out and rotating it so that the unit (1,0) of the second plane lies on top of (a,B) [and (0,0) on (0,0)], then (c,d) will lie on top of the product.

What is the point of denying the existence of numbers in consistent number systems? Are you looking for an excuse to avoid learning the mathematics?

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First, speaking as moderator, please include enough context in your posts to make clear to whom or to what you are responding.

If it were actually the case that infinite numbers "do not exist", then I would expect the assumption that they do exist which is regularly used by mathematicians to lead to a contradiction.  But it has not, as far as I know.

If the quote marks surrounding "do not exist" were meant to quote someone's actual words, I cannot find anywhere in this thread where those words were said. If they were meant as scare quotes, meaning something you would like to distance yourself from, then you really should be more specific. There are physical existents and mental existents, meaning things that exist in the physical world and things that are mental constructs. The notion of the infinite as a physical existent is an invalid notion, primarily because it contradicts the law of identity. To physically exist means to have physical limits on attributes; identity is existence.

We can distinguish from physical existents those mental existents that are a product of consciousness. The infinite as a product of consciousness does not have a referent in physical reality; it is a concept of method, in the purely psychological sense. The infinite as a concept of method refers to the potential of a given method -- a means of representing a process -- not an actual physical existent.

It is important to distinguish these two cases -- a physical existent from a mental existent -- lest it lead to confusion.

As for complex numbers, these are as real as points on a plane.  Consider x+iy to be the point (x,y)....

The geometric representation in the complex plane has been well-known for more than two centuries, but unless you make clear in what sense "real" is meant, its use here can be equivocal. A graphical representation is just that -- a representation-- and in that sense it is certainly real. But it is not real in the physical sense.

What is the point of denying the existence of numbers in consistent number systems?  Are you looking for an excuse to avoid learning the mathematics?

I am not sure whom you are referring to here, but, speaking as moderator, on THE FORUM we do not permit speculation as to member's motives. Please confine your remarks to the ideas, not to the person. If you are unclear about our policies, please review the Guidelines for THE FORUM. Thanks.

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Excuse me, my previous message was messed up by the emoticons.

Incidentally, the way to avoid the emoticon problem with "B)" showing as an emotion when it was meant otherwise, is to uncheck the "Enable emoticons?" just below the text box when preparing the message.

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