1=0.999... infinities and box of chocolates..Phliosophy of Math...

[Undefined]

Hi Yazata.

Both Leibniz and Newton would have disagreed with you guys.

They would have said that 0.999... + an infinitesimal = 1

where an infinitesimal is an infinitely small number.

Put another way, 1 - 0.999... = an infinitesimal

Many mathematicians weren't comfortable with the idea of infinitesimals, which despite its Newtonian and Leibnizian pedigree seemed more intuitive than mathematically rigorous.

Weierstrauss rigorously reformulated the foundations of calculus in terms of limits in the 19'th century. That seems to me to be where the idea might have originated that 0.999... = 1, since the limit of 0.999... is one. At any rate, most mathematicians seem to have kind of uncritically assumed that infinitesimals were finished and that they were of historical interest at best.

Interestingly, in the 1960's Robinson produced a rigorous mathematical account of infinitesimals. That led to an alternative formulation of the foundations of calculus in terms of infinitesimals, called non-standard analysis. (And yes, non-standard analysis is part of 'mainstream mathematics' and isn't the least bit crankish.)

So it seems to me that this thread really raises an interesting and perhaps rather important issue in the philosophy of mathematics. It's not an appropriate occasion for insulting people.

Excellently observed, mate! Kudos.

Especially when, since then, all the work on the nature/effect of Fractal Equations/concepts, Quantum Uncertainty and Chaotic Systems steps/tunneling states etc have increased the background understandings of the 'infinitesimal of effectiveness' in reality, even if the mathematics has yet to be advanced to the completeness stage where it can recognize and treat this 'infinitesimal' concept more in tune with the contextual reality requirements for 'making sense' in reality as well as in its own purely limited abstract axiomatic construct/context.

Well done in posting those most pertinent and interesting observations of yours, mate!

 

[Quantum Quack]

Ahhh, yes. You are right, QQ, when you read it like that. I took it to mean that 'reality' should not be brought into a 'mathematical' issue such as the one mentioned. That is why I made the point that even those who would use that excuse to 'evade' the implications in any context (real or maths) are now using a 'reality' process (see Tach's example I pointed to) as a counter-argument...which has done nothing except CONFIRM the validity of MD's insistence on reality meanings when attempting to explore the actual philosophical basis FOR all the abstract maths assumptions which these discussions now question.

Sorry to you and to someguy1 if I wasn't clear that it was Tach's use of reality based arguments even as he disparaged MD's reality based approach!

Thanks for the assist in clarifying that, QQ. Good man! Cheers.

You are not the only one Undefined, who's thinking and analytical abilities are disturbed by Tach's attitude.. It is indeed why he does it to achieve just thioihhcuyghj purpose.. gosh see what I mean...:bawl:
At least you are man enough to admit a mistake and apologize where appropriate...takes courage and fortitude...
I have other people here in Melbourne reading these posts, who are also ending up "scrambled" by his attitude.. so you are not alone...

 

[Fednis48]

Both Leibniz and Newton would have disagreed with you guys.

They would have said that 0.999... + an infinitesimal = 1

where an infinitesimal is an infinitely small number.

Put another way, 1 - 0.999... = an infinitesimal

Many mathematicians weren't comfortable with the idea of infinitesimals, which despite its Newtonian and Leibnizian pedigree seemed more intuitive than mathematically rigorous.

Weierstrauss rigorously reformulated the foundations of calculus in terms of limits in the 19'th century. That seems to me to be where the idea might have originated that 0.999... = 1, since the limit of 0.999... is one. At any rate, most mathematicians seem to have kind of uncritically assumed that infinitesimals were finished and that they were of historical interest at best.

Interestingly, in the 1960's Robinson produced a rigorous mathematical account of infinitesimals. That led to an alternative formulation of the foundations of calculus in terms of infinitesimals, called non-standard analysis. (And yes, non-standard analysis is part of 'mainstream mathematics' and isn't the least bit crankish.)

So it seems to me that this thread really raises an interesting and perhaps rather important issue in the philosophy of mathematics. It's not an appropriate occasion for insulting people.

This is an interesting take on things. I guess I hadn't thought about the metaphysical status of infinitesimals. By definition, any infinitesimal is smaller than any arbitrary number we can choose to compare it with. So if a mathematical expression does contain an infinitesimal, throwing the infinitesimal away will always leave the value of the expression unchanged up to arbitrary precision. Since math is just a language for formalized logic anyway, I don't worry too much about whether throwing away the infinitesimal gives a correct description of what the number "really is"; insofar as the results are the same, I'm inclined to go with the simpler method, which is not to bother tracking infinitesimals. But it's interesting (and probably grants insight into some fundamental properties of arithmetic) that one can build math in a non-standard way using infinitesimals.

That said, I'm a little disappointed that people seem receptive to Motor Daddy's arguments. Finer points of math philosophy aside, the idea that an object cannot be divided into three equal parts is utter nonsense. As gmilam alluded to back on page 2, the non-terminating decimal representation of 1/3 is just an artifact of the base-10 system we use. In a base-9 system, 1/3=0.3 (not repeating). Fundamentally, there is no reason why 1/3 is any less precise of a number than 1/2, or any other fraction. It might be a matter of mathematical formalism whether 1 and 0.(9) represent the same number of two numbers that are infinitesimally different from one another, but it's important to keep in mind that at most this is a question of labeling. Trying to extend this reasoning to argue against the reality of certain fractions is beyond inane.

 

[Quantum Quack]

This is an interesting take on things. I guess I hadn't thought about the metaphysical status of infinitesimals. By definition, any infinitesimal is smaller than any arbitrary number we can choose to compare it with. So if a mathematical expression does contain an infinitesimal, throwing the infinitesimal away will always leave the value of the expression unchanged up to arbitrary precision. Since math is just a language for formalized logic anyway, I don't worry too much about whether throwing away the infinitesimal gives a correct description of what the number "really is"; insofar as the results are the same, I'm inclined to go with the simpler method, which is not to bother tracking infinitesimals. But it's interesting (and probably grants insight into some fundamental properties of arithmetic) that one can build math in a non-standard way using infinitesimals.

That said, I'm a little disappointed that people seem receptive to Motor Daddy's arguments. Finer points of math philosophy aside, the idea that an object cannot be divided into three equal parts is utter nonsense. As gmilam alluded to back on page 2, the non-terminating decimal representation of 1/3 is just an artifact of the base-10 system we use. In a base-9 system, 1/3=0.3 (not repeating). Fundamentally, there is no reason why 1/3 is any less precise of a number than 1/2, or any other fraction. It might be a matter of mathematical formalism whether 1 and 0.(9) represent the same number of two numbers that are infinitesimally different from one another, but it's important to keep in mind that at most this is a question of labeling. Trying to extend this reasoning to argue against the reality of certain fractions is beyond inane.

An interesting and probably easy question to answer comes to mind... is

0.999.. (or equivalent) = 1 in other bases.
In other words is the treatment of infinity similar for other bases..
in particular one that is of interest to me being base 12. [duodecimal]

 

[Pete]

An interesting and probably easy question to answer comes to mind... is

0.999.. (or equivalent) = 1 in other bases.
In other words is the treatment of infinity similar for other bases..
in particular one that is of interest to me being base 12. [duodecimal]

I believe this has been answered before, and it's certainly addressed on the Wikipedia page.
In base 2, 0.111... = 1
In base 3, 0.222... = 1
In base 4, 0.333... = 1
... and so on and so forth.

 

[Billy T]

...In base 2, 0.111... = 1
In base 3, 0.222... = 1
In base 4, 0.333... = 1
... and so on and so forth.

No one has taken up this challenge below. Why?

{end of my post 335in the 1 = 0.999... thread} ...PS, a puzzle for the mathematically skilled or inclined:
Using only the integer 1 thur 1000, what is the longest repeat period possible in their infinite rational decimal equivalent?
"Repeat period" is for example 4 in 0.12341241234... but that may not equal the ratio of any two integers given the requirement that both numbers in the ratio are no more than 1000.

For the really advanced mathematicians: Still using only the first 1000 number, is there a base (not 10) in which an even longer repeat period exists?

If because it is too hard, then do it for integers less than 100 (or your choice, perhaps 64?) instead of 1000.

 

[Pete]

997 is the highest prime <1000.
I think x/997 (x is any number under 1000 except 997) has 996 repeating digits in any base (except 997 and its multiples)

 

[Billy T]

997 is the highest prime <1000.
I think x/997 (x is any number under 1000 except 997) has 996 repeating digits in any base (except 997 and its multiples)

Nice observation, but question was in which base is the "repeat interval" the longest and is there no other rational integer ratio with a longer repeat interval?

Later by edit:

Oh, now I see your point. 996 is very likely the correct answer. I did not know that recipicals of primes have repeat lengths one less than the prime. Is there a proof of that?

 

[Pete]

There is no ratio of integers under 1000 with a repeat interval more than 996, in any base.

I think that 1/997, 2/997, 3/997, ... 996/997 all have repeat interval of 996 in any base, (except base 997, of course).

 

[Tach]

An interesting and probably easy question to answer comes to mind... is

0.999.. (or equivalent) = 1 in other bases.

The equality is preserved in all bases. If $$x_{(n)}=y_{(n)}$$ in base (n) then $$x_{(p)}=y_{(p)}$$ in base (p).

 

[Undefined]

Good morning, Fednid48, Yazata, QQ, MD, Pete, Tach, everyone.

This is an interesting take on things. I guess I hadn't thought about the metaphysical status of infinitesimals. By definition, any infinitesimal is smaller than any arbitrary number we can choose to compare it with. So if a mathematical expression does contain an infinitesimal, throwing the infinitesimal away will always leave the value of the expression unchanged up to arbitrary precision. Since math is just a language for formalized logic anyway, I don't worry too much about whether throwing away the infinitesimal gives a correct description of what the number "really is"; insofar as the results are the same, I'm inclined to go with the simpler method, which is not to bother tracking infinitesimals. But it's interesting (and probably grants insight into some fundamental properties of arithmetic) that one can build math in a non-standard way using infinitesimals.

Your honest (as usual; Kudos!) self-observation allows that you had not before considered (at least not to the extent that I and some others here apparently have) the deeply important and comprehensively instructive "META-PHYSICAL status" infinitesimals. By implication, it is possible that you may also had not considered (again, at least not to the extent that I and some others here apparently have) the even more reality-pertinent 'REAL-PHYSICAL status" of the logically deducible and physically recognized "infinitesimal of effectiveness" which is inherent in the understandings and dynamical basis of QM "a 'something' tunneling through 'infinitesimal nothings' to 're-produce' that 'same something' elsewhere", as observed; of CHAOS THEORY "infinitesimal steps from starting simplicity towards infinite complexity", as observed; and, of FRACTAL MATHEMATICS "iteration equations/effects based on fractal infinitesimal variability between iterations", again, as observed.

So maybe we should, as you put it, "...worry..." about the infinitesimal in all its contexts, Fednis48?

Especially if our common goal is 'completeness and cross-consistency' between all abstract mathematical and concrete physical 'systems of thought/modeling'?

Again, assuming our collective ultimate aim IS to actually address THE reality and not just keep playing with NON-reality. I think humanity has grown up and left the basement of 'video games' and is now beginning to actually 'face the real world' outside the basement 'virtual world' disconnect with what's really important in the final analysis of what science is about.

That said, I'm a little disappointed that people seem receptive to Motor Daddy's arguments. Finer points of math philosophy aside, the idea that an object cannot be divided into three equal parts is utter nonsense. As gmilam alluded to back on page 2, the non-terminating decimal representation of 1/3 is just an artifact of the base-10 system we use. In a base-9 system, 1/3=0.3 (not repeating). Fundamentally, there is no reason why 1/3 is any less precise of a number than 1/2, or any other fraction. It might be a matter of mathematical formalism whether 1 and 0.(9) represent the same number of two numbers that are infinitesimally different from one another, but it's important to keep in mind that at most this is a question of labeling. Trying to extend this reasoning to argue against the reality of certain fractions is beyond inane.

Again, you seem to miss that the only counter-argument so far, against MD's REALITY-based approach to that 1/3 (ie, "divide something into 3 equal parts IN REALITY"-----whatever abstract number system you wish to play with in UN-reality) has been offered by Tach , using EXACTLY the same REALITY-based approach which he had 'derided' MD for using!

Here is Tach's supposed 'counter' example, effectively constituting a real 'division exercise' on a real thing (read MD's "pie" for Tach's "circle"):

Really? In 8-th grade they teach you how to inscribe an equilateral triangle in a circle. This means that you either haven't taken that class yet or that you flunked it.

I don't think that you understood the simple exercise. You are given a compass and a ruler. Draw a circle. Divide the circumference of the circle in 3 equal parts using the two tools given to you. You have 10 minutes from when this post is active. If you cannot do it , you flunked 8-th grade geometry. Live with it.

See, Fednis48? There is no REAL 'proof' of the a-priori/abstract assumption by Tach (and now effectively repeated by yourself above) that any 'real division' scenario results in 3 'equal' parts instead of MD's perfectly valid 'real context' observation/argument (even using Tach's OWN example) that IN REALITY there is NO basis for making a-priori/abstract ASSUMPTION that the 'division' CAN give 3 'equal' parts AT ALL in that particular scenario.

Of course, IF we FIRST construct or 'compose', something shown to BE the SUM of PRIOR objects which INITIALLY SUPPLIED the 3 "equal' parts to some 'composite object' MADE from those 3 "equal' objects in the first instance, then it would be trivial to 'reverse' that operation and 'decompose' it into 3 equal parts. HOWEVER, since the 'pie/circle' in this particular context has NOT PREVIOUSLY BEEN 'composed' using 3 'equal' parts in the first place, THEN can be NO A-PRIORI/ABSTRACT ASSUMPTION that it can NOW, in reality, BE 'decomposed' into 3 "equal" parts. It cannot BE 'proven' in that case where no 'composition' case has been demonstrated to arrive AT the 'pie/circle' REAL OBJECT we want to 'divide' via 1/3 in REALITY, as per MD's point made so far.( * )

Hence the math-versus-reality perspectives 'impasse'; and where the 'last step' infinitesimal-of-effectiveness 'non-zero difference' comes in to save the day; which would make MD's (and others, including myself) observe that always (whatever number system/abstraction one plays with), in reality, the expression 1/3 represents/results in:

At least ONE of those three parts being slightly (by an unavoidable infinitesimal of effectiveness) greater than the other TWO parts.



Anyhow, this discussion should highlight yet again the possible dangers of letting our 'mathematics' rule us blindly in all contexts, since it tends to depart from reality more and more as UN-real abstraction/assumption is piled upon UN-real abstraction/assumption.

Yes, Mathematics is useful, but let's not let it run away with itself and us, and so insidiously OBSCURING from our ken more and more that reality which we are striving to elucidate for REAL and not just for VIRTUAL. Yes?

Let's NOW actually "...worry about..." and really consider properly and exhaustively all the contextual aspects/effects/meanings etc OF that "infinitesimal of effectiveness' LAST REAL STEP between something and zero/balance/singularity etc etc states which occur in reality but which bamboozle our mathematics because, as axiomatically defined so far, the maths gives infinities and singularities when it breaks down and our current equations 'blows up' to indicate the end of its 'domain of applicability' boundary conditions which it cannot handle with any reality sense result.

Thanks for your time and trouble in making your own very interesting contribution to this PHILOSOPHY of MATHS thread, Fednis48, Yazata, MD, everyone. Much appreciated; and by more than just Quantum Quack, I assure you!

Great thread, Quantum Quack; Kudos for starting it and setting just the right tone for polite and insightful discourse roght from the start! Cheers!


( * )This is a subtle but extremely important 'contextual reality' aspect requiring careful consideration to understand/discuss these things properly. For example, we can in the first instance 'compose' a '6' from three "equal" parts of '2'; and its reverse is trivially achieved by dividing precisely into its original three "equal parts of '2'. HOWEVER, we CANNOT DEMONSTRABLY IN REALITY INITIALLY 'compose' directly a REAL "UNIT WHOLE" object (a UNIT WHOLE 'pie', a UNIT WHOLE 'circle') which is NOT ALREADY amenable to being 'composed' FROM three "equal" parts in the first place. Unless anyone can INITIALLY MAKE a REAL, WHOLE UNIT 'pie' or 'circle' FROM THREE REAL WHOLE UNITS which can BE demonstrated in reality to BE "equal" to each other, then no amount of starting from the 'other end' can PROVABLY (not abstractly/assumedly) 'derive' three "equal" parts in reality (again, not abstract/assumed) division operation applied to those reality cases.

 

[Fednis48]

997 is the highest prime <1000.
I think x/997 (x is any number under 1000 except 997) has 996 repeating digits in any base (except 997 and its multiples)

Based on this answer, it sounds like you have a theorem that says "1/n has n-1 repeating digits if n is a prime number." Is that true? If so, do you have a link to an easy-to-understand proof? If not, how did you come up with this answer?

 

[Fednis48]

Your honest (as usual; Kudos!) self-observation allows that you had not before considered (at least not to the extent that I and some others here apparently have) the deeply important and comprehensively instructive "META-PHYSICAL status" infinitesimals.

Let's not go overboard here. Calling the metaphysical status of infinitesimals "deeply important and comprehensively instructive" is your words, not mine. Like I tried to say above, mathematics is an extremely refined formalism for deductive logic, nothing more. The fact that including or discarding infinitesimals can lead to two constructions that give the same results is interesting, and it probably sheds light on certain fundamental aspects of logic. But to ask whether infinitesimals (or any other mathematical constructs, for that matter) are "real" is the wrong question, in my opinion.

By implication, it is possible that you may also had not considered (again, at least not to the extent that I and some others here apparently have) the even more reality-pertinent 'REAL-PHYSICAL status" of the logically deducible and physically recognized "infinitesimal of effectiveness" which is inherent in the understandings and dynamical basis of QM "a 'something' tunneling through 'infinitesimal nothings' to 're-produce' that 'same something' elsewhere", as observed; of CHAOS THEORY "infinitesimal steps from starting simplicity towards infinite complexity", as observed; and, of FRACTAL MATHEMATICS "iteration equations/effects based on fractal infinitesimal variability between iterations", again, as observed.

... what? I'm sorry, but none of these examples even make sense. Quantum particles don't "tunnel through infinitesimal nothings." Chaos theory doesn't have anything to do with infinitesimal steps, and the equations that produce fractals produce finite changes with every iteration.

Especially if our common goal is 'completeness and cross-consistency' between all abstract mathematical and concrete physical 'systems of thought/modeling'?

Again, assuming our collective ultimate aim IS to actually address THE reality and not just keep playing with NON-reality. I think humanity has grown up and left the basement of 'video games' and is now beginning to actually 'face the real world' outside the basement 'virtual world' disconnect with what's really important in the final analysis of what science is about.

Math helps us describe reality. It is not the same as reality. Insofar as we can refine our mathematics to make better predictions about reality, I'm all for it. But I say worrying about the existence of infinitesimals, which by their definition do not make any finite difference in our predictions, is a waste of time.

See, Fednis48? There is no REAL 'proof' of the a-priori/abstract assumption by Tach (and now effectively repeated by yourself above) that any 'real division' scenario results in 3 'equal' parts instead of MD's perfectly valid 'real context' observation/argument (even using Tach's OWN example) that IN REALITY there is NO basis for making a-priori/abstract ASSUMPTION that the 'division' CAN give 3 'equal' parts AT ALL in that particular scenario.

Well, ok. Let me ask you this: can you really divide an object into two equal parts? If not, then you're making a VERY bold claim, and I'd be willing to debate it with you. But it sounded to me like MD was saying division into three equal parts specifically was impossible because 1/3 is a non-terminating decimal. That claim would be indisputably wrong; 1/3 is only non-terminating because we do math in base 10, and reality cannot depend on our choice of base.

Let's NOW actually "...worry about..." and really consider properly and exhaustively all the contextual aspects/effects/meanings etc OF that "infinitesimal of effectiveness' LAST REAL STEP between something and zero/balance/singularity etc etc states which occur in reality but which bamboozle our mathematics because, as axiomatically defined so far, the maths gives infinities and singularities when it breaks down and our current equations 'blows up' to indicate the end of its 'domain of applicability' boundary conditions which it cannot handle with any reality sense result.

I'll put it this way. Show me a self-consistent formulation of math that makes meaningfully different predictions by including infinitesimals, and I'll examine it with interest. Until then, it seems to me that you're just reifying logical abstractions for no reason other than that their absence makes you uncomfortable.

 

[Tach]

Again, you seem to miss that the only counter-argument so far, against MD's REALITY-based approach to that 1/3 (ie, "divide something into 3 equal parts IN REALITY"-----whatever abstract number system you wish to play with in UN-reality) has been offered by Tach , using EXACTLY the same REALITY-based approach which he had 'derided' MD for using!

You seem to have a lot of difficulty with this simple problem of geometry.

Here is Tach's supposed 'counter' example, effectively constituting a real 'division exercise' on a real thing (read MD's "pie" for Tach's "circle"):

Have you managed to find the solution? It is part of 8-th grade geometry curriculum.

See, Fednis48? There is no REAL 'proof' of the a-priori/abstract assumption by Tach (and now effectively repeated by yourself above) that any 'real division' scenario results in 3 'equal' parts instead of MD's perfectly valid 'real context' observation/argument (even using Tach's OWN example) that IN REALITY there is NO basis for making a-priori/abstract ASSUMPTION that the 'division' CAN give 3 'equal' parts AT ALL in that particular scenario.

Whatever the word salad above, the simple fact is that they teach you in 8-th grade geometry how to divide the circle circumference with a ruler and acompass. repeating MD's mistakes doesn't make your posts right, makes them fringe.

Of course, IF we FIRST construct or 'compose', something shown to BE the SUM of PRIOR objects which INITIALLY SUPPLIED the 3 "equal' parts to some 'composite object' MADE from those 3 "equal' objects in the first instance, then it would be trivial to 'reverse' that operation and 'decompose' it into 3 equal parts. HOWVER, since the 'pie/circle' in this particular context has NOT PREVIOUSLY BEEN 'compose' using 3 'equal' parts in the first place, THEN NO A-PRIORI/ABSTRACT ASSUMPTION that is can NOW, in reality, BE 'decomposed' into 3 "equal" parts.

You should stop spreading gross falsities like the above.

It cannot BE 'proven' in that case where no 'composition' case has been demonstrated to arrive AT the 'pie/circle' REAL OBJECT we want to 'divide' via 1/3 in REALITY, as per MD's point made so far.

False. You should really stop spreading anti-science, this simple exercise has been solved more than 2000 years ago.

 

[rpenner]

997 is the highest prime <1000.
I think x/997 (x is any number under 1000 except 997) has 996 repeating digits in any base (except 997 and its multiples)

An interesting hypothesis, however I will guess that you are wrong. Let's see if I can prove it in less than 30 minutes. The time now is 3:37.

 

[rpenner]

997 is the highest prime <1000.
I think x/997 (x is any number under 1000 except 997) has 996 repeating digits in any base (except 997 and its multiples)

An interesting hypothesis, however I will guess that you are wrong. Let's see if I can prove it in less than 30 minutes. The time now is 3:37.

Period 2 in base 996.

$$\frac{1}{997} = \frac{0 \times 996 \; + \; 995}{996^2} \; + \; \frac{1}{997} \times \frac{1}{996^2}$$
The time is 3:42 (most of which was typesetting)

Likewise the period is 83 in base 9, 12, 16 and others. Example:
$$\frac{1}{997} = \frac{8775328885789415945325986869077718617264471136983801149150534027722980939616352751147495193673235}{16^{83}} \; + \; \frac{1}{997} \times \frac{1}{16^{83}}$$

Likewise the period is 12 in bases 91, 252 and perhaps others.

And now (4:14) -- I have automated the process.

http://www.wolframalpha.com/input/?...[2,996],+MultiplicativeOrder[#,+997]+<13+&+]]

The period is 1 in bases 998, 1995
The period is 2 in bases 996, 1993
The period is 3 in bases 304, 692, 1301, 1689
The period is 4 in bases 161, 836, 1158, 1833
The period is 6 in bases 305, 693, 1302, 1690
The period is 12 in bases 91, 252, 745, 906, 1088, 1249, 1742, 1903

 

[Billy T]

An interesting hypothesis, however I will guess that you are wrong. Let's see if I can prove it in less than 30 minutes. The time now is 3:37.

I asked for proof that 1/p where p is a prime has repeat length of p-1back in post 108, but know it is not always true. i.e. 1/5 =0.2 but it does seem true for 1/7 = 0.142857 142857 142857 Also note 1/11 =0.909090909090909... repeat length of 2, not 10, but blocks of ten do repeat.

1/13 =0.076923 076923 076923 ... has repeat length of 6 not 12, but again blocks of 12 do repeat. Seem to be something interesting going on here.
The puzzle I proposed long ago is more interesting than I realized.

As 1/5 = 0.2000 0000 0000 .... one could say there is a repeat length of 4 (but of any integer number also).

When the prime, p, is a factor of the base, that may be a special case?

 

[Quantum Quack]

@undefined

HOWVER, since the 'pie/circle' in this particular context has NOT PREVIOUSLY BEEN 'compose' using 3 'equal' parts in the first place, THEN NO A-PRIORI/ABSTRACT ASSUMPTION that is can NOW, in reality, BE 'decomposed' into 3 "equal" parts.

I think what you are getting at is the paradoxical nature of applying infinity on a finite object. If I read you correctly you are I feel quite correct in stating as you have. If not then I apologize.

Example by way of problem:
take a house brick and divide it into an infinite number of slices so that all slices are equal in thickness.

Q: How thick are the slices?

Then :

Q:How many slices does it require to recompile the brick?

Now if the slice thickness is deemed to exist then the slices have a finite thickness.
How many finite thick slices are needed to recompile the brick?

Choices: 1] a finite number of slices or 2] an infinite number of slices?

If the slices are deemed to be "finite" infinitesimals, or given a fixed value, then the recompiling the brick is a finite function and not the same infinite function that was used to de-compile the brick.

Compare with using 2 dimensional slices instead.

What do you discover from the thought experiments?

The bottom line question is:
If you divide a house brick into an infinite number of slices :
Do the resultant slices exist in 3 dimensional space or not. If so, in what way and with what dimension of thickness are they?
To me this highlights the paradox associated with real world use of "Infinity"
1/infinity = 0 or does it equal 1/infinity

 

[Quantum Quack]

Its exactly the same question being asked of

1- 0.999... =
zero
or
1/infinity
when applied to the real 3 dimensional world.

if the answer is 0 then the brick vanishes.. non-existent. [and can not be recompiled as (0 x infinity) = 0]
if the answer is 1/infinity
what happens?
Does the brick [slices compiled] still exist?

see the paradox?
When:
1/infinity = 0
"When infinitely reducing a sphere, the sphere ceases to exist as a sphere" and it is a one way street only, for once the sphere is reduced infinitely it can not be recompiled from nihilo"

When:
1/infinity = 1/infinity
"When infinitely reducing a sphere, the sphere maintains form as a sphere" and it can be recompiled from 1/infinity" ~ yet this grants 1/infinity a finite value

 

[Undefined]

Hi Fednis48.

Thanks for your considered and, as usual, courteous reply.

Let's not go overboard here. Calling the metaphysical status of infinitesimals "deeply important and comprehensively instructive" is your words, not mine. Like I tried to say above, mathematics is an extremely refined formalism for deductive logic, nothing more. The fact that including or discarding infinitesimals can lead to two constructions that give the same results is interesting, and it probably sheds light on certain fundamental aspects of logic. But to ask whether infinitesimals (or any other mathematical constructs, for that matter) are "real" is the wrong question, in my opinion.

Yes, that "deeply important and comprehensively instructive" was my opinion of it in the context of overall discussion OF all its 'meanings' to whomever/whatever. Sorry if my construction was unclear and so inadvertently implied the opinion was in any way yours also on that. My bad!

Yes, that other point you make about mathematics is already a 'given' here (at least as far as I am concerned). However, this is a PHILOSOPHY of MATHS discussion, and it is the PRE-mathematics process which leads TO the current mathematics axioms/formalism that I am approaching all these points from. I already understood where you were coming from, and I made further comment based on where I am coming from. That's all. I didn't mean for it to sound like I didn't recognize the current maths formalism for what it is. Again, sorry if my posts gave any other impression, mate!

... what? I'm sorry, but none of these examples even make sense. Quantum particles don't "tunnel through infinitesimal nothings." Chaos theory doesn't have anything to do with infinitesimal steps, and the equations that produce fractals produce finite changes with every iteration.

According to Standard Model assumptions/postulates/theory, the Big Bang Scenario is essentially an UNIDENTIFIED IN REALITY some fundamental (infinitesimal?) 'nothing' state/thing which due to 'quantum fluctuation' of/in that 'nothing' became 'something'?

In my reality based perspective, it is an Energy-space arena 'something' that existed all along, but we will not go into that now.

Insofar as Standard Model goes, the space-time and all its mathematically modeled 'properties' ultimately depend on certain 'infinitesimal' things associated with 'pointlike' things which when treated in some 'mathematical space treatments' effectively represent the 'forces' and 'energy' and other dynamical entities which are used for the abstractions of reality into the mathematical formalism dependent 'models'.

So I now ask:

If 'pointlike' (ie, having no real or abstract dimensional extent) things are NOT ultimately THE closest thing (logically and effectively in reality context) TO 'infinitesimals of reality effectiveness' (regardless of mathematical formalism abstractions/modeling), then what are these pointlike 'nothings' that underlie all theorizing in 'from-nothing-to-something' models/treatments like that used for the conventional Big Bang Standard Model? Fair enough question, mate?

Math helps us describe reality. It is not the same as reality. Insofar as we can refine our mathematics to make better predictions about reality, I'm all for it. But I say worrying about the existence of infinitesimals, which by their definition do not make any finite difference in our predictions, is a waste of time.

Again, the 'reality limitations' of mathematics AS IT STANDS NOW is not disputed, so it is a 'given' whenever such philosophical discussions OF it (and of other 'limited' systems of thought, especially including the physics system) are brought into 'review' like this via NON-prejudicial and 'uncommitted stance' discussions which are not yet FORMAL PRESENTATIONS as 'completed review'. So a certain leeway and patience needs to be allowed for developing all the various thoughts and observations which such philosophical discussions will inevitably elicit for further discussion in the various contexts they arise in. Yes?

To emphasize: this thread is obviously designed to encourage 'philosophical stage' discussion of many things, one of which happens to (inescapably) be the status of the conventional mathematics as so far 'developed' towards (hopefully) the ultimate goal of 'completeness' in every sense, which IF it does become complete in every sense, then must eventually BECOME CAPABLE OF reflecting THE reality in a cross-contextual (abstract math-concrete reality) way that satisfies everyone on all sides.

Well, ok. Let me ask you this: can you really divide an object into two equal parts? If not, then you're making a VERY bold claim, and I'd be willing to debate it with you. But it sounded to me like MD was saying division into three equal parts specifically was impossible because 1/3 is a non-terminating decimal. That claim would be indisputably wrong; 1/3 is only non-terminating because we do math in base 10, and reality cannot depend on our choice of base.

Within the limit of accuracy of measuring 'parts', we can. The philosophic/reality question which MD and others raise has to to with the actual reality capability----EVEN GIVEN SUCH ACCURACY----to divide one WHOLE UNIT into 3 "equal" parts UNLESS we already prove that the particular "whole unit" was actually composed of such 3 "equal" parts IN THE FIRST place. That is the tricky/subtle thing. No mathematical abstraction of 1/3 process can be proven tom result in 3 "equal" parts unless it has already been proven that it can be...and that has yet to be proven in reality with the real 'pie/circle' UNIT WHOLE we start with in MD's/Tach's reality-based example.

Can you prove that the pie/circle unit whole object contains three equal parts before you actually divide it in reality? If so, then you just prove a triviality for THAT and suchlike cases, and not necessarily for ALL reality cases. Hence where the reality context "infinitesimal of effectiveness" could come in and make constitute some 'bridging axiom/object/process' which makes the mathematics become more reality reflective WITHOUT necessarily having to ditch its other excellent useful axioms/results. That is where I am coming from. That's all.

I'll put it this way. Show me a self-consistent formulation of math that makes meaningfully different predictions by including infinitesimals, and I'll examine it with interest. Until then, it seems to me that you're just reifying logical abstractions for no reason other than that their absence makes you uncomfortable.

This is a tentative INFORMAL stage philosophy of maths (and other tricky/subtle things) thread. What you ask there is not yet advisable, since conclusion/completeness OF these and other like discussions elsewhere have to cover much more ground to develop greater common ground before that FORMAL stage is reached for all these unconventional forays into the subjects/issues in question. It would help get us further more quickly if some 'proofs' from 'outside' the current maths formalisms could be presented to 'counter' the issues/points raised so far?

Anyhow, thanks again, Fednis48, for your excellently considered and courteously expressed, and always most interesting and evenhanded, contributions to these discussions and others. I think I may speak for everyone here when I say that your posts are invariably (except by the case of trolls, of course (*) ) appreciated by all here! Cheers, mate!



(*)Speaking of trolls, all non-troll members please be aware that, as per excellent admin/mod advice, I am effectively ignoring a certain troll at this time. Thanks.