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Author Topic:   Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
Jon
Inactive Member


Message 1 of 24 (544226)
01-24-2010 8:36 PM


Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
In the thread "0.99999~ = 1 ?" (Message 1), talk has turned somewhat to a topic slightly different from that first introduced and proposed. Specifically, the issue at point is no longer the distinctness of two numbers, but the foundations of the entire system of Mathematics (what I called MATHSYSTEM) in relation to the real world (what I called REALWORLDSYSTEM).

As I stated in Message 87

... more a matter of definition than a matter of real-world fact.

A consequence of some proofs in Message 110:

P · "All #s with property Z are REAL within MATHSYSTEM (i.e., ((R/Z)/M) (M=MATHSYSTEM)"
P · "0.9999| has property Z"
P · "(The REAL of 0.9999| is True) given MATHSYSTEM is True (i.e., ((R/0.9999|)/M))"
P · "[proof of M=True]"
C · "(R (the REAL of 0.9999|) is True... period"

Proof that .9999| is not DISTINCT (from 1):
P · "It is false that 0.9999| is both REAL and DISTINCT (from 1) (~(R·D))"
P · "(R (the REAL of 0.9999|) is True... period"
C · "0.9999| is not DISTINCT (from 1)"

... which was a consequence of the proofs explained in Message 89:

Proof that .9999| is REAL:
P · [P1 for .9999| being a REAL number]
P · [P2 for .9999| being a REAL number]
etc.
C · [Conclusion that .9999| is a REAL number]

Proof that .9999| is not DISTINCT (from 1):
P1 · "It is true that 0.9999| is REAL"
P2 · "It is true that 0.9999| is DISTINCT (from 1)"
P3 · "It is false that both P1 and P2 are true (~(P1·P2))"
P4 · "[conclusion from proof of 0.9999| being a REAL number], i.e., P1 is true"
C · "P2 is false, i.e., .9999| is not DISTINCT (from 1)"

... along with the definition of REAL I had been given as being ultimately dependent on the MATHSYSTEM (see posts in reply to me, especially by Dr. A.), and as RAZD concurred in Message 117:

RAZD writes:

math does not need to conform to reality

... the MATHSYSTEM appears to have no necessary relationship to the REALWORLDSYSTEM. As was shown in my reply to Dr. A, which was Message 120:

1 = 3/3 = 1/3 + 2/3 = 0.3333| + 0.6666| = 0.9999| ≠ 1

So, the MATHSYSTEM introduces a function that equates 0.9999| with 1 and thereby closes the paradoxical loophole, such that we may continue using the MATHSYSTEM as a representation of the REALWORLDSYSTEM. Of course, just because we can mend one system so that it will represent another, does not indicate a necessary representative property of the former system in regards the latter, i.e., it does not show that the former system represents by necessity the latter system, but merely shows that it represents it, not necessarily by necessity.

Afterall, any system that represented necessarily another system would not need a function to close paradoxical loopholes, as such loopholes would not exist. The fact that the MATHSYSTEM has introduced such a function in an attempt to represent the REALWORLDSYSTEM, shows that the MATHSYSTEM is not necessarily representative of the REALWORLDSYSTEM.

... were there a necessary relationship between the MATHSYSTEM and the REALWORLDSYSTEM, they would both paradoxically collapse in upon themselves—I mean, the paradox related to their existences would cause them to cease existing.

In Message 123, Rrhain writes in response to my question whether there is any reality in the real numbers:

Rrhain writes:

That is a philosophical question, not a mathematical one. You are playing to the distinction between Platonists and non-Platonists.

And that, to most mathematicians:

... the objects that mathematics studies are real.

So, I would like to challenge any mathematicians who hold this view to support such. I am not convinced that the MATHSYSTEM is necessarily linked to the reality that it describes. As I pointed out in Message 120, certain internal operators of the MATHSYSTEM which function to tweak its failures to match the REALWORLDSYSTEM so it can be a better describer are part of the evidence I will offer initially that the MATHSYSTEM could not be a necessary representative of the REALWORLDSYSTEM, i.e., it does not represent it by necessity, but rather by convention. Or, in the words of Rrhain, I'm going to pull a Bert.

Jon

Edited by Jon, : No reason given.


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Replies to this message:
 Message 3 by Dr Adequate, posted 01-27-2010 10:52 AM Jon has responded
 Message 6 by nwr, posted 01-27-2010 2:59 PM Jon has responded
 Message 15 by Son Goku, posted 01-28-2010 5:51 AM Jon has not yet responded

  
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Message 2 of 24 (544571)
01-27-2010 8:12 AM


Thread Copied from Proposed New Topics Forum

    
Dr Adequate
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Posts: 16107
Joined: 07-20-2006
Member Rating: 7.7


Message 3 of 24 (544580)
01-27-2010 10:52 AM
Reply to: Message 1 by Jon
01-24-2010 8:36 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
Your decision to write in a language of your own invention makes your point unnecessarily obscure.

However, in plain English we may note that it is trivially the case that the proposition that a mathematical structure forms a good model for some aspect of reality is necessarily a scientific theory to be confirmed or disconfirmed by observation and experiment; a question which is extrinsic to the mathematics as such.

Whether or not you find this observation helpful depends, of course, on what the heck it is that you're talking about.


This message is a reply to:
 Message 1 by Jon, posted 01-24-2010 8:36 PM Jon has responded

Replies to this message:
 Message 4 by Jon, posted 01-27-2010 2:27 PM Dr Adequate has responded

  
Jon
Inactive Member


Message 4 of 24 (544616)
01-27-2010 2:27 PM
Reply to: Message 3 by Dr Adequate
01-27-2010 10:52 AM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
Dr. A writes:

we may note that it is trivially the case that the proposition that a mathematical structure forms a good model for some aspect of reality is necessarily a scientific theory to be confirmed or disconfirmed by observation and experiment; a question which is extrinsic to the mathematics as such.

Ahh, and it is the fact that Mathematics—like a scientific theory composed of words in their structure, the units of Language—is falsifiable—like a scientific theory composed of words in their structure, like any Linguistic proposition, i.e., a proposition which relies on Linguistic modes for its conveyance—that shows the arbitrarily disconnected, conventional (i.e., not by necessity) link that Math has with Reality. In an Empirical epistemology1, Reality2—and all things which follow by necessity from it—is unfalsifiable. Thus Mathematics—like Language—, being falsifiable, cannot be said—within an Empirical epistemology—to follow necessarily from Reality. And if it is the case that it does not follow by necessity, then any 'following' it appears to do is not part of the epistemological certainty that is Reality within an Empirical epistemology and is therefore not 'knowable' and true but merely an illusion or coincidence, or, in the case of Math, a result of agreed-upon convention.

Now, of course, one could hold to a Mathematical epistemology, in which numbers and their operators were the only things real, with all else being illusory, from the ground below to the very Self. In such a case—really any case—the link between Math and Reality would be completely unnecessary3 to even exist let alone be consistent. One may also attempt to hold to both a Mathematical and Empirical epistemology with the condition that one would have to show the two epistemologies to be consistent both within each other and in relation to each other else be caught in a contradiction, or discard the entire pursuit of the epistemologically certain altogether.

So, as far as Math and Reality go, our options are rather limited to:
1. Deny the possibility of a successful resolution to the pursuit of knowledge.
2. Show the cross-epistemological consistency of Math and Empiricism.
3. Accept Math as real and Reality as illusory.
4. Accept Reality as real and Math as illusory.

Choosing 1 is silly, whether correct or not, it gets us nowhere; choosing 3 or 4 is fine as each one is perfectly consistent within itself, but also admits to a non-necessary link between Math and Reality; choosing 2 requires evidence of cross-consistency. Obviously I stand at 4, maintaining Math's link to Reality to be conventional; this thread is for folk who stand at 2 to back up that stance.

Jon
__________
1 I use this in a broad sense to mean what one accepts as 'real', i.e., what one can be certain is 'knowable' versus what may or may not be 'knowable' and therefore may or may not be 'real', with 'knowable' being loosely equivalent to 'epistemological certainty'. In this sense, anything that is 'real' and thus 'knowable' cannot be falsified, else it is not 'real'. An Empirical epistemology, then, would say that the natural world (Reality, see foot note 2) exists in such a form as to be unfalsifiable.
2 Simply meaning the natural world as opposed to other worlds, any of which may be epistemologically real (lower case) whether part of Reality (the natural world, upper case) or not.
3 Recall, actual existence says nothing of necessary existence; the link may still exist, just not necessarily so.


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 3 by Dr Adequate, posted 01-27-2010 10:52 AM Dr Adequate has responded

Replies to this message:
 Message 5 by New Cat's Eye, posted 01-27-2010 2:48 PM Jon has responded
 Message 12 by Dr Adequate, posted 01-27-2010 4:33 PM Jon has not yet responded

  
New Cat's Eye
Inactive Member


Message 5 of 24 (544620)
01-27-2010 2:48 PM
Reply to: Message 4 by Jon
01-27-2010 2:27 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
__________

Did you know that if you type </hr>, it becomes a:


?


This message is a reply to:
 Message 4 by Jon, posted 01-27-2010 2:27 PM Jon has responded

Replies to this message:
 Message 7 by Jon, posted 01-27-2010 3:00 PM New Cat's Eye has responded

  
nwr
Member
Posts: 5586
From: Geneva, Illinois
Joined: 08-08-2005


Message 6 of 24 (544621)
01-27-2010 2:59 PM
Reply to: Message 1 by Jon
01-24-2010 8:36 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
Specifically, the issue at point is ... the foundations of the entire system of Mathematics (what I called MATHSYSTEM) in relation to the real world (what I called REALWORLDSYSTEM).

That's not really an issue. Most mathematicians will tell you that mathematics is not about the real world.

You seem to be confused by the fact that mathematicians sometimes say that their mathematics is real. However, as Rrhain tried to point out, that's more of a philosophical issue.

The major philosophies of mathematics are:

  • Platonism: mathematical objects exist in a world of ideal platonic forms (sometimes loosely described as "Plato's heaven".

  • Realism (or mathematical realism): this is just another name for platonism.

  • Fictionalism: mathematical objects (such as numbers) are useful fictions.

  • Intuitionism: mathematical objects exist only in the intuitions of people (particular, of mathematicians).

  • Constructivism: mathematics is a science of what we can construct mathematically.

  • Formalism: mathematicians make meaningless formal marks, and play rule-based games with these marks.

In practice, most mathematicians are platonists. A few are fictionalists, but you would probably have difficulty distinguishing between platonists and fictionalists. constructivists and intuitionists have much in common. In particular, they tend to be skeptical of much that platonists say about infinite sets. I am not convinced that there are any actual formalists, so formalism is mostly a fall back position used to explain to skeptics (such as you) what it is that mathematicians do.

I consider myself a fictionalist.

I have never asked an intuitionist about whether 0.9999... is equal to 1. That's partly because I don't actually know any intuitionists in real life, though I know of some. My best guess is that intuitionists and constructivists would agree that 0.9999... is equal to 1. That they are equal is a matter of well accepted convention.


This message is a reply to:
 Message 1 by Jon, posted 01-24-2010 8:36 PM Jon has responded

Replies to this message:
 Message 8 by cavediver, posted 01-27-2010 3:06 PM nwr has responded
 Message 11 by Jon, posted 01-27-2010 4:22 PM nwr has responded
 Message 14 by Iblis, posted 01-27-2010 8:50 PM nwr has acknowledged this reply

  
Jon
Inactive Member


Message 7 of 24 (544622)
01-27-2010 3:00 PM
Reply to: Message 5 by New Cat's Eye
01-27-2010 2:48 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
__________

Did you know that if you type </hr>, it becomes a:


?

Of course; but then my footnotes become as separated from the main text as the technical 'who has replied' information on the bottom. I want them set off from the main text, not cut off from it.

Anyway, off topic, please no more on the decision to use underscores in place of line codes.

Jon


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 5 by New Cat's Eye, posted 01-27-2010 2:48 PM New Cat's Eye has responded

Replies to this message:
 Message 10 by New Cat's Eye, posted 01-27-2010 3:35 PM Jon has acknowledged this reply

  
cavediver
Member (Idle past 1935 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 8 of 24 (544623)
01-27-2010 3:06 PM
Reply to: Message 6 by nwr
01-27-2010 2:59 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
My best guess is that intuitionists and constructivists would agree that 0.9999... is equal to 1. That they are equal is a matter of well accepted convention.

My constructivist friend denies the existence of the limit implied by .9999~, but accepts .9999~ symbollically as a representation of 1. Similary with .33333~ and 1/3, etc.

ABE: sorry, how rude. Good to see you around

Edited by cavediver, : No reason given.


This message is a reply to:
 Message 6 by nwr, posted 01-27-2010 2:59 PM nwr has responded

Replies to this message:
 Message 9 by nwr, posted 01-27-2010 3:18 PM cavediver has not yet responded

  
nwr
Member
Posts: 5586
From: Geneva, Illinois
Joined: 08-08-2005


Message 9 of 24 (544624)
01-27-2010 3:18 PM
Reply to: Message 8 by cavediver
01-27-2010 3:06 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality
My constructivist friend denies the existence of the limit implied by .9999~, but accepts .9999~ symbollically as a representation of 1.

Interesting. Thanks.

ABE:

ABE: sorry, how rude. Good to see you around

No big deal, and not rude at all. (I asked for reinstatement so I could comment in this thread).

Edited by nwr, : add comment on edited change in the post to which this was responding.


This message is a reply to:
 Message 8 by cavediver, posted 01-27-2010 3:06 PM cavediver has not yet responded

Replies to this message:
 Message 16 by Jazzns, posted 01-28-2010 11:01 AM nwr has responded

  
New Cat's Eye
Inactive Member


Message 10 of 24 (544626)
01-27-2010 3:35 PM
Reply to: Message 7 by Jon
01-27-2010 3:00 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
Of course

Excuse me for fogetting just how frickin' brilliant you are

but then my footnotes become as separated from the main text as the technical 'who has replied' information on the bottom. I want them set off from the main text, not cut off from it.

To each his own, I guess.

Anyway, off topic, please no more on the decision to use underscores in place of line codes.

Who do you think you are?

Gosh, a simple "Thanks, yeah." would have sufficed... and not make you look like such a pompous douchebag.


This message is a reply to:
 Message 7 by Jon, posted 01-27-2010 3:00 PM Jon has acknowledged this reply

  
Jon
Inactive Member


Message 11 of 24 (544632)
01-27-2010 4:22 PM
Reply to: Message 6 by nwr
01-27-2010 2:59 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
nwr writes:

You seem to be confused by the fact that mathematicians sometimes say that their mathematics is real. However, as Rrhain tried to point out, that's more of a philosophical issue.

Well, I realize that 'real' numbers are a specific type of numbers within the MATHSYSTEM which do not necessarily have any relation to the 'real' in Reality. My main point in everything has been that 0.9999| is a Mathematical concept which does not exist in the Real world, likewise for 1. Instead, there is a Reality for which Math has two modes of representation, which it must then equate through rules of its own making.

And yes, it is very much philosophical. My point has been rather simple all along; some folk, however, have been confusing what I say and muddling it up so much that I needed to start a separate thread to try to explain my take: "I do not think Math has any necessary relationship to Reality, and I use the 0.9999| example to show it. As far as the Mathematical equivalence of 0.9999| to 1, I think it is True. As far as for their Reality-based equivalence, I think it, like them, does not exist."

Thank you for your informative post. I guess by those standards I too would be a Fictionalist/Intuitionalist. Not sure who could ever claim to be a Formalist, obviously the Math symbols have meaning, even if to just one person.

Thanks,
Jon


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 6 by nwr, posted 01-27-2010 2:59 PM nwr has responded

Replies to this message:
 Message 13 by nwr, posted 01-27-2010 5:19 PM Jon has not yet responded

  
Dr Adequate
Member
Posts: 16107
Joined: 07-20-2006
Member Rating: 7.7


Message 12 of 24 (544633)
01-27-2010 4:33 PM
Reply to: Message 4 by Jon
01-27-2010 2:27 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
Ahh, and it is the fact that Mathematics—like a scientific theory composed of words in their structure, the units of Language—is falsifiable—

No.

like a scientific theory composed of words in their structure, like any Linguistic proposition, i.e., a proposition which relies on Linguistic modes for its conveyance—that shows the arbitrarily disconnected, conventional (i.e., not by necessity) link that Math has with Reality.

The relationship between math and reality isn't conventional, it's discovered. The fact that two apples plus three apples is five apples isn't a mere social agreement like driving on the right. The isomorphism between reality and the structure of the natural numbers actually exists.


This message is a reply to:
 Message 4 by Jon, posted 01-27-2010 2:27 PM Jon has not yet responded

  
nwr
Member
Posts: 5586
From: Geneva, Illinois
Joined: 08-08-2005


(1)
Message 13 of 24 (544640)
01-27-2010 5:19 PM
Reply to: Message 11 by Jon
01-27-2010 4:22 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
I guess by those standards I too would be a Fictionalist/Intuitionalist.

You fit more as a constructivist or intuitionist. The intuitionists take the view that there are no infinite decimal expansions, only finite ones. Fictionalists don't have a problem dealing with fictions, so most would not have a problem with infinite decimal expansions.

In the particular case of 0.9999..., what we have is a recurring decimal expansion. And that's different from an infinite decimal expansion. We can actually have a notation for recurring decimals that only requires writing down finitely many symbols. While a platonist would take that notation as a shorthand for an infinite decimal expansion, a constructivist or intuitionist might take it as a separate notation in its own right which does not require anything infinite. I think that's what cavediver's constructivist friend was implying (see Message 8).


This message is a reply to:
 Message 11 by Jon, posted 01-27-2010 4:22 PM Jon has not yet responded

  
Iblis
Member (Idle past 2187 days)
Posts: 663
Joined: 11-17-2005


Message 14 of 24 (544683)
01-27-2010 8:50 PM
Reply to: Message 6 by nwr
01-27-2010 2:59 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
I have never asked an intuitionist about whether 0.9999... is equal to 1.

I've answered already though, at Message 81. It is. To me, it's like asking whether 1 - 0 - 0 - 0 ... is somehow different from 1. I cannot understand most of the arguments from the other side, they seem to be trying to say that one divided by infinity is something other than 0.

I don't have any problem with repeating decimals, they are simply a weird effect of the base system. I do have a problem with non-repeating decimals, but it is not the same problem the constructivists apparently have.

Interestingly, CatholicScientist gives a variation on the same proof that I do. Message 7


This message is a reply to:
 Message 6 by nwr, posted 01-27-2010 2:59 PM nwr has acknowledged this reply

  
Son Goku
Member
Posts: 1152
From: Ireland
Joined: 07-16-2005


(1)
Message 15 of 24 (544753)
01-28-2010 5:51 AM
Reply to: Message 1 by Jon
01-24-2010 8:36 PM


Re: Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Real
Hey Jon,

nwr has already outlined most of the mathematical philosophies, I just wanted to comment on a two things. First of all, as far as I can gather a lot of mathematicians take the view that mathematics is real in the same way that chess is real.
Specifically,
Regardless of whether you come up with the axioms of the system (rules of chess) or discover them, you choose to study the system (choose to play chess). However within the system, most mathematicians think that the "proof paths" are real. I'll explain what I mean in an analogy:
Given the rules of chess the Fool's Mate always existed as a possible play sequence in the game. It didn't come into existence when somebody played it for the first time. It was always a possible outcome of this particular game.
Similarly you may come up with the axioms of complex numbers, but in that mathematical system it was "always" true that all polynomials had a solution (fundamental theorem of algebra). There's no way you could have invented the complex numbers and not have the fundamental theorem of algebra. Nobody (except I think the most extreme formalists) objects to this.

Finally from a physicists point of view, it turns out (as was essentially said by Dr Adequate), that some of these mathematical systems do in fact correspond to reality or match reality to a large degree. The counting of discrete objects is matched by the functioning of the natural numbers.

In the case of 0.999... = 1, this identity is a property of the Real numbers (not their only strange property). However it is properties like this which allow the real numbers to support calculus, which matches the real world.*

*I should say that most constructionists that I have met (i.e. all two of them) believe that 0.999.. = 1, the transcendentals and all other wierd Real number stuff, is just unfortunate formal junk we have to put up with to obtain calculus. That is they agree with calculus, they just don't like the bizarre number system it's based on. To quote Hermann Weyl: (paraphrasing)
"[Caculus is] A solid house built on a foundation of quicksand."

P.S. Nice to see you again nwr!

Edited by Son Goku, : Greeting


This message is a reply to:
 Message 1 by Jon, posted 01-24-2010 8:36 PM Jon has not yet responded

Replies to this message:
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