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

Wow, a simple glitch/loop in the decimal representation of thirds generates 18 pages of discussion! No wonder I always found philosophy to be boring.

MD said:
What is "immediate" about .999...? You have a REPEATING DECIMAL, so the operation is continuously repeating and never complete, rate of repeat being irrelevant!
What operation? You're thinking of the steps used in long division, as I said. This is how it's done, you think. There is only one way to do it, you think.

Why do you think that?

What operation? You're thinking of the steps used in long division, as I said. This is how it's done, you think. There is only one way to do it, you think.

Why do you think that?

I think it repeats because of the ... and the fact that it's referred to as a repeating decimal. What do you think the term "repeating" means? Why does it repeat? How does something repeat instantaneously?

Thanks for your comments. You are much better versed in formal mathematics than I (a physicist) ever was but I do have a question:

If rational decimal, RD = 0.ab.... where a&b are integers, perhaps equal as in 0.3333... or not as in RD =0.121212... is it less well established from axioms that 10RD =a.babab.... than that terminating decimal, TD =0.ab00000... has 10TD = a.b0000...? Or are they equally well built on the axioms?
They are equally well built on axioms as are all valid inferences.

I only said I "multiplied" to more clearly explain my general procedure but as note earlier post, my moving of the decimal points I think is well defined and FUNDAMENTAL in the meaning of the notational system used.
To the formalist there is no "fundamental" without axioms. To the formalist the excursion towards "fundamental" is all timey-wimey twaddle until you establish a ground floor by stating your axioms. I agree with you that the necessary equivalence between 10×3.1415... and 31.415... is easy to conceptualize as a necessary consequence of our base-ten number system, but proving their equivalence requires talking about infinite, something that mere arithmetic of rationals is ill-suited to do.

I.e. I believe if 0.abc = 0.def then a.bc = d.ef and both =10x0.abc or 10x 0.def just from what the base 10 notational system means. (First place to left of decimal point tells how many units of one, second to the left tells how many units of ten, etc. and etc. for the right side of the decimal point)

Do you agree?
Of course I agree, as this follows from finite addition of terms and is provable specifically or generally for all finite numbers of terms via finite induction. Arithmetic of rational numbers alone, however, does not give us an axiom system strong enough to address the case when the number of terms is larger than any finite number.

Also do you agree that multiplication is only defined for multiplication by integers as multiplying M by n is defined as adding M to itself n times. The multiplying M by some non-integer is defined from an algorithm known to be valid for multiplying by an integer or IFF at least one of M & n is and integer with help of the commutative law (I think that is the name) which for ordinary math states: AxB = BxA.*
That's Peano multiplication. In the real numbers the definition of multiplication has to be defined in terms that make sense with whatever is adopted in place of the axiom of completion. If it is Cauchy completion, then $$\sqrt{2} \, \cdot \, \pi$$ is defined as the limit of the sequence you get by multiplying pairwise corresponding terms of any sequences of rational numbers that converge to $$\sqrt{2}$$ and $$\pi$$, respectively. Thus the theorem that $$\sqrt{2} \, \cdot \, \pi = \pi \, \cdot \, \sqrt{2}$$ in the reals is a theorem proven from the commutative law of the rationals (which itself can be proven from the commutative law of natural numbers which may be axiomatic or rest on the Peano definition of multiplication of natural numbers).

E.g. 3x7.5 is not computed directly from the definitions of Mxn but makes use of this commutative law first to get 7.5 x 3 as you can add 7.5 up 3 times but not add 3 up with itself 7.5 times but 3.4 x7.5 can not be computed directly from the definition of multiplying. We can only assume (or define) the result via some algorithm known to be valid by it producing the same results as the products the definition can be applied to.
This procedure has to be handled for negative numbers (how do you add -2 to itself -5 times?), for rational numbers and the reals, so the concept is understood. But the actual details seem to differ from how you are thinking of them because we abandon the Peano axioms as soon as we allow negative numbers. Thus we bootstrap ourselves natural numbers -> integers -> rational numbers -> real numbers -> complex numbers and algebraic completeness.

What do you think the term "repeating" means? Why does it repeat? How does something repeat instantaneously?
The decimal is a repeating pattern of digits, the pattern is otherwise known as the period. In the case of 1/3, the period is 1.
But that does not mean the division operation repeats, and 1/3 certainly implies a single (i.e. not repeating) operation. The numbers are distinct from any operation on them.

Since we've been discussing the properties of the number 1, how many times can you multiply or divide 1 by itself? Is 1 x 1 = 1 or 1 x 1 x 1 = 1 "immediate", or do you have to do in steps?

Wow, a simple glitch/loop in the decimal representation of thirds generates 18 pages of discussion!

That's just this thread. It's been going on for several threads before that, in multiple places on the board.

No wonder I always found philosophy to be boring.

I'm not convinced that this is a philosophy thread. It's more of a mathematics thread, but it isn't that either. It's mostly an ego and attitude thread, I guess. Whatever it is, it's plopped itself down in philosophy like a cookoo's egg.

I love philosophy with an absolute passion. But mathematics? Not nearly so much. This thread has succeeded in one thing, reminding me of why that is.

$$(\frac{1}{\2}=.5)$$ and $$(.5*2=1.0)$$
$$(\frac{1}{\4}=.25)$$ and $$(.25*4=1.0)$$
$$(\frac{1}{\5}=.2)$$ and $$(.2*5=1.0)$$

So it appears to me, Tach, that according to your concept of the above, zero times infinity should equal one according to you.

$$(\frac{1}{\infty}=0)$$ and $$(0*\infty=1)$$

Is that what you are saying, Tach, that since $$\frac{1}{\infty}=0$$ then it stands to reason that $$(0*\infty=1)$$???

Good, you pass 3-rd grade arithmetic.

Nope, basic calculus says that $$0* \infty=undetermined$$

Yep

Nope, you fail. Again.

Your basic error is that $$\frac{1}{\infty} =0$$ does not imply $$0 * \infty=1$$. Math is tough.
So then calculus must think $$undetermined* \infty=0$$, right???
Nope $$undetermined* \infty=undetermined$$. Math is really tough. The fact that you'll never learn it should start sinking in.

I have been observing the above exchange, and it again highlights clearly the incompleteness and insufficiency of the relevant axioms when the excuse of 'undetermined/undefined' must be invoked because the axioms cannot treat the consequent logic-flow 'results' in a complete and consistent manner.

It is the symptom of the gap between the maths Axioms and the reality Physics, which is at the heart of the problem for the mathematics, which produces these 'undetermined/undefined' outcomes (like the one pointed out by MD above) whenever the maths/axioms are pushed to the limit of their competence/reliability.

That is why we need the overhaul of the maths axioms to reflect the reasonableness of Physical reality FIRST, and the Maths exercise SECOND. Good luck trying to make sense from the current INCOMPLETE and (as MD has highlighted in the above exchange) patently inadequate axiomatic formulation, guys!

Hi arfa brane, everyone.

The decimal is a repeating pattern of digits, the pattern is otherwise known as the period. In the case of 1/3, the period is 1.
But that does not mean the division operation repeats, and 1/3 certainly implies a single (i.e. not repeating) operation. The numbers are distinct from any operation on them.

Since we've been discussing the properties of the number 1, how many times can you multiply or divide 1 by itself? Is 1 x 1 = 1 or 1 x 1 x 1 = 1 "immediate", or do you have to do in steps?

arfa, I have already more than once, cautioned against trivial self-selecting 'examples' which are not 'proofs' at all, but merely trivial 'composition' which is equally trivially 'decomposed'.

For example, using greater-than-unit numbers (like 9/3) is not in the spirit of proving the FRACTIONAL argument about 1/3, is it?

And now above, you offer an equally trivial 1-to-1 'example' for your argument about 'operation'.

Using such trivial and self-selecting 'devices' is tantamount to using a NON-action, rather than a valid OPERATION.

The operation is essentially an ACTION, not a hypothetical one-off conclusion/assumption as to result.

So, guys, let's keep the examples and arguments NON-trivial. Let's reflect the REALITY of ACTION as opposed to some PHILOSOPHICAL INSTANTANEOUS state which has been assumed a-priori, hey?

Let's all try to keep:

- the fractional discussion FRACTIONAL, and not unity/over-unity. Yes?

- the operational discussion OPERATIONAL, and not NON-action or trivial reversibility of neutral arguments/constructs. Yes?

Thanks. Cheers, arfa, MD, everyone.

Undefined said:
Using such trivial and self-selecting 'devices' is tantamount to using a NON-action, rather than a valid OPERATION.
Operations are defined on sets, like the integers.

For numbers n,m each an integer, addition is defined like this:

m + n = n + m

Which says nothing at all about "how" to add two integers together, but does say the operation of addition is commutative.
Multiplication is also commutative in the integers: m x n = n x m.
Nowhere does this say you add together n copies of m, or m copies of n. But that's ok, you just define m x n = n + n + ... + n, m times; usually you see this expressed like: $$\sum_{k=1}^m n$$
The operation is essentially an ACTION, not a hypothetical one-off conclusion/assumption as to result
No idea what you mean by this.

I have been observing the above exchange, and it again highlights clearly the incompleteness and insufficiency of the relevant axioms when the excuse of 'undetermined/undefined' must be invoked because the axioms cannot treat the consequent logic-flow 'results' in a complete and consistent manner.

There is no "incompleteness". There is no "insufficiency".

It is the symptom of the gap between the maths Axioms and the reality Physics, which is at the heart of the problem for the mathematics,

The issue being discussed has nothing to do with physics, it is an elementary math problem. This has been explained to you countless times.

which produces these 'undetermined/undefined' outcomes (like the one pointed out by MD above) whenever the maths/axioms are pushed to the limit of their competence/reliability.

I am thrilled to see that you and MD are in agreement while sharing fringe misconceptions. Mainstream scientists, not so much. The math as we know it doesn't suffer from any issues and it is fully reliable, contrary to your persistent fringe dronings.

That is why we need the overhaul of the maths axioms to reflect the reasonableness of Physical reality FIRST, and the Maths exercise SECOND.

We need no such things, your theories belong in "Alternate Theories". Current axioms are firmly established.

Good luck trying to make sense from the current INCOMPLETE and (as MD has highlighted in the above exchange) patently inadequate axiomatic formulation, guys!

Most of us can make perfect sense of the math, why can't you?

There is no "incompleteness". There is no "insufficiency".

The issue being discussed has nothing to do with physics, it is an elementary math problem. This has been explained to you countless times.

I am thrilled to see that you and MD are in agreement while sharing fringe misconceptions. Mainstream scientists, not so much. The math as we know it doesn't suffer from any issues and it is fully reliable, contrary to your persistent fringe dronings.

We need no such things, your theories belong in "Alternate Theories". Current axioms are firmly established.

Most of us can make perfect sense of the math, why can't you?

Your 'explanation' to MD was "undetermined".

And now you say "Current axioms are firmly established".

So, those current axioms are established so well that an 'answer' of "undetermined" is OK with you as a "result" of those axioms?

Not all of us are so 'accepting' of such inadequacy/insufficiency in our maths axioms.

The only way to remedy those axioms is to make them based on more real rather than abstract foundations.

You, of course, are welcome to stay put in your perfectly abstract but insufficient axiomatic 'world' where "undetermined" is an OK 'result' from said current axioms.

Fortunately for the advancement of the maths and the science, not everyone is as satisfied and willing as you seem to be to 'settle' for such INcomplete and INsufficient maths/axioms and physics/postulates as those currently extant.

I think the bottom line with this thread is that mathematics refuses to acknowledge that a paradox exists when considering infinities.
the question :
what does 1/infinity = ?
and according to one PHD:
strictly speaking, 1 / infinity is infinitesimal (very small), and 1 / undefined is zero. that is the language when talking about Pure Mathematics. However, in applied sciences, where approximations are often prevalent, 1 / infinity is considered as zero, and the word "undefined" is seldom used.
what this tells us is that when a mathematician reckons that 1/infinity = 0 then he is talking in approximations.
What it also tells us is that the only way 0.999... can be calculated to equal 1 is by using the approximation of 1/infinity=0
Other wise 0.999... = 1 is defined arbitrarily by the use of limits.

The problem is simply that mathematics can not cope with a paradox and 1/infinity is a paradox when related to zero.
@Yazata
One of the main reasons for starting this thread was to allow philosophical math discussion on how mathematics appears to still have problems after 1000's of years when attempting to deal adequately, in an absolutely exact way, with paradoxes.
And this is a classic case in question.

Which is why I introduced the image below to aid in furthering that discussion.

and the next image to further consolidate the point [excuse the pun]

Logically one can not find the infinitesimal unless one expands from zero and not contracts from 1
Draw a "ball" that has the smallest possible diameter greater than zero.
Then ask what is in side that 3 dimensional ball?

What seems to be missed or over looked when talking about a ball that is 1/infinity in diameter is that it clearly demonstrates an infinitesimal opening into zero dimensionality in the real 3 dimensional space universe.
Now when talking about quantum entanglement and how two 1/2 particles can communicate instantaneously over vast distances one can see potential for a solid explanation for F.T.L. communications.
As the zero dimensional space "implied" can be used to facilitate such communications.

Quantum Quavk said:
I think the bottom line with this thread is that mathematics refuses to acknowledge that a paradox exists when considering infinities.
How long would you say mathematics has refused to acknowledge this paradox? Since a few centuries B.C,, since the 19th century, or just since this thread started?
Now when talking about quantum entanglement and how two 1/2 particles can communicate instantaneously over vast distances one can see potential for a solid explanation for F.T.L. communications.
They don't communicate instantaneously, there is no FTL communication.

They don't communicate instantaneously, there is no FTL communication.

eh?
I suppose you failed to read or agree with what the Phd Guy said.. that in pure mathematics 1/infinity =/=0
strictly speaking, 1 / infinity is infinitesimal (very small), and 1 / undefined is zero. that is the language when talking about Pure Mathematics. However, in applied sciences, where approximations are often prevalent, 1 / infinity is considered as zero, and the word "undefined" is seldom used.

@arfa_brane,
How long would you say mathematics has refused to acknowledge this paradox? Since a few centuries B.C,, since the 19th century, or just since this thread started?
actually a fair jab..
maybe my tainting every one with the Tach brush is a tad unjustified..

Quantum Quack said:
eh?
I suppose you failed to read or agree with what the Phd Guy said.. that in pure mathematics 1/infinity =/=0
Well, I think I'll stick with the idea that infinity is defined in a non context-free way, That is to say, it depends on the context.
But what do you think a mathematical definition of infinity has to do with communication? What's the context?

Well, I think I'll stick with the idea that infinity is defined in a non context-free way, That is to say, it depends on the context.
by all means... as you wish...
But what do you think a mathematical definition of infinity has to do with communication? What's the context?
If zero in the physical world is defined as being a point that is as described by the reduction of a ball to the diameter of 1/infinity [ which is not zero ] it means that the zero point exists in a three dimensional sense even though it is zero dimensional. [zee paradox!]

The distance between two 1/2 particles in 3 dimensional space may be a positive value yet simultaneously zero as well.

This suggest that a possible mechanism for quantum entanglement phenomena may rest with this possible understanding of the zero point, it also further indicates a path towards understanding how energy can be derived from zero dimensions as well.

So this issue of 1/infinity =0 or =/= 0 is not trivial in it's potential ramifications.
one of which is: If the above is possible then all points in 3 dimensional space have a zero point potential, which means that zero space and 3 d space co-exist universally.

ok.....for you sci fi buffs,, hyper space would be a common term.
"....to infinity and beyond..."
and Qm's entanglements are only the tip of the iceberg

Your 'explanation' to MD was "undetermined".

The FACT is that basic calculus teaches that $$0*\infty=undetermined$$. This doesn't mean that the math behind the statement is in need of any "corrections", especially from people like you.

And now you say "Current axioms are firmly established".

Right. And there is no contradiction.

So, those current axioms are established so well that an 'answer' of "undetermined" is OK with you as a "result" of those axioms?

Yep, you need to learn because you are obviously ignorant on the subject.

Not all of us are so 'accepting' of such inadequacy/insufficiency in our maths axioms.

The only way to remedy those axioms is to make them based on more real rather than abstract foundations.

Not by you, that is for sure, you don't know even the basics.

You, of course, are welcome to stay put in your perfectly abstract but insufficient axiomatic 'world' where "undetermined" is an OK 'result' from said current axioms.

Yep, undetermined is the correct answer. It may not be for the fringers but it is perfectly understood by the people who are math educated. Yoy should try taking a class sometimes, get educated.

Those who claim that 0.9999... is not 1.0 or true are (1) illogical

No, no, no, no. Just the opposite. If you review the last few pages of the thread, you will see that it's the heretics who are trying to invent informal logical arguments for their (possibly erroneous) position. Their opponents are more apt to resort to insults and putdowns. (The heretics are 'ignorant'. they are 'fringers', they 'aren't educated', they 'need to take a class'.) That kind of stuff communicates disdain very well, but not ideas. Or there are remarks like 'it's INFINITE', or 'you're thinking sequentially'. Those remarks may or may not be true, but they aren't logical arguments.

1. In several posts I wrote something along the lines of this: A decimal point followed by a string of 9's of some arbitrary finite length n does not equal 1.0. Adding another 9 to the string, so that we have (n+1) 9's, won't make it equal 1.0 either. So why can't we argue, by mathematical induction, that no string of 9's, no matter how long it's extended, will ever equal 1.0?

That might well be false, perhaps for some elementary reason, perhaps for some more subtle and philosophically interesting reason. The thing is, it certainly seems to be reasonable on its face.

2. Motordaddy argued that if we divide a whole into 2 equal parts, we have 2 1/2's, which equal the whole. If we divide a whole into 947 equal parts, we have 947 1/947's, which equal the whole.

But if, as is being asserted, 1/infinity = 0, then dividing a whole into an infinite number of parts will make it disappear entirely. Both Motordaddy's and my own intuitions disagree, thinking that what we would have instead is an infinite number of infinitely small parts, which would still equal the whole.

That may indeed be false, but it doesn't seem to be illogical. My impression is that Isaac Newton and Gottfried Leibniz might have agreed with Motordaddy and myself, which suggests that even if we're all wrong, these aren't totally stupid ideas to entertain.

or (3) have assumed different axioms than common ones

I don't think that any of us are arguing axiomatically. But yeah, we probably are employing different presuppositions. That's not necessarily a bad thing. It might be a very interesting thing.

I'm most emphatically not a mathematician. But my layman's impression of the history of mathematics suggests that despite both Newton and Leibniz making use of infinitesimals, many mathematicians weren't comfortable with them, because infinitesimals seemed to have no place in number theory and threatened to create difficulties in proof theory and whatever.

Two hundred years later, in the 19th century, Weierstrass produced a mathematically sound way to redefine things in terms of limits. That seemed to have the laudable result of eliminating the poorly defined and mathematically questionable infinitesimals. So, for more than a century, calculus teaching made every attempt to keep students from thinking in terms of infinitesimals, even though (in fact precisely because) infinitesimals seemed so intuitive to so many students. I think that we might be seeing continuing blow-back from that history here in this thread, as people repeat their own teachers' disdain for infinitesimals.

But starting maybe fifty years ago, things started to change. Infinitesimals made a startling comeback. Non-standard analysis appeared, which despite its name was kind of a revival of the older thinking, supported by all kinds of new consistency proofs and stuff. And it even proved fruitful, leading to the development of interesting new areas of mathematics such as the theory of hyperreal numbers, where infinitesimals finally found their home.

So... while the technical details of all this stuff are WAY above my pay-grade, my impression is that BOTH sides in this little argument might conceivably be right. It's possible to define things in terms of limits, Weierstrass-style, and that's consistent and it works. And my impression is that it's ALSO possible to conceive things in ways in which infinitesimals appear. Names like Abraham Robinson are associated with that and it's consistent and it works too, despite both approaches seemingly being inconsistent with one another.

The 0.999... = 1.0 argument might conceivably be an example of that inconsistency.

I've actually seen introductory calculus textbooks that take a non-standard infinitesimals approach. I don't believe that they are in widespread use, but apparently their authors thought that the alternative approach might make more intuitive sense to their beginning students.

or (3) poorly understand the common base-10 notational system.

Seeing as how Sciforums is a layman's board, that's almost a certainty.

or are (4) just internet trolls.

I think that a few people here do have some rather crankish ideas. This is Sciforums, that's always going to be true, in any thread. But others of us are basically saying that what the 0.999... = 1.0 people are asserting seems counterintuitive and illogical.

That doesn't necessarily mean wrong. Many ideas in mathematics and science are initially counterintuitive. What's needed in these cases is a convincing account of why the counterintuitive ideas really do make sense.

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