x = 0.999...
10x = 9.999... — Michael

Nice. But it still begs the question: what does it mean to say x = 0.999...?
It means an infinity of 9s but what can that mean when infinity is not a number?
You have to say x = lim 9/10 + 9/100 + 9/1000 + ...
And we are back to square 1. (Uh, I mean square 0.999...)

How do you know it's infinity and not, say, an octillion? — InPitzotl

Because I know it is not any nameable number.

The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1. — fdrake

Note that in the article cited in the op they don't write

$\sum_{i=1}^{\infty}$ etc = x

They write

lim $\sum_{i=1}^{\infty}$ etc = x

These are two different concepts.

If you don't understand these issues, you should read through jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you. — fdrake

I have read through it. These are mathematical expressions and as such they are symbols. They represent infinity. But mathematicians were aware of these issues when formulating the calculus and they cautioned against saying 'equals'. They said we should say 'Tends towards the limit'

What is 1/3 in decimal? — Michael

Infinitesimals have never really been understood rigorously. Have you heard of Berkeley's "Ghosts of departed quantities"? Below 'Fluxions' means infinitesimals.

Ghosts of departed quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities (Boyer 1991), Berkeley wrote:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?[6]

Edwards describes this as the most memorable point of the book (Edwards 1994). Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions. (Katz & Sherry 2012)

Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).

Which is exactly why you write 0.999... — fdrake

Yes, I understand what you are saying. But if infinity is not a number how can you have an infinity "of"?

0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1. — fdrake

.999... could not be the limit. To write the limit you'd have to have .999999999999999999999999999999999999999999999999999 - an infinity of 9s. And we can't write that, whatever it means.

.999... is a symbol for 'an infinity of' 9s. But what does that mean?

What is 1/3 in decimal? — Michael

I have no idea!!! I suspect there is 'an infinity of 3s' but what does that mean? That's the crux of the biscuit.

So, how many numbers are there? — InPitzotl

I don't know because I don't know if 'how many' applies to infinity. At the beginning of the theory of limits mathematicians were careful to say that we should say a series 'tends towards' a limit. It is a conservative statement.

Then you don't know what the symbols mean and should read the OP's article! — fdrake

But does anybody know? Intuitively yes, we can see that the limit is 1. But limit is not the same as equals. The argument is subtle. What is being said is 'After an infinity of 9s'. That is what I am suspicious about. I'm not sure what 'an infinity of' means. Or if it is a coherent statement.

That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...} — fdrake

I don't think so. .999... describes the series. The limit of it is infinitely far away.

Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same. — jorndoe

But the problem is what does .999... mean? How many 9s are we talking about? An infinity of them, of course. But what is an infinity 'of' something?

Actually it does, that's why they use the equals sign. It's the entire essence of calculus. — Pantagruel

True. But they don't mean .999... = 1. They mean the Limit of the infinite sum = 1. There's a difference.
The sum being 9/10 + 9/100 + 9/1000 + ...

because that’s just what decimal notation means. And since the limit of the series of partial sums of that infinite series is 1, that means the total sum of that infinite series represented by 0.999... is also 1, so 0.999... = 1. — Pfhorrest

But it has to do with the way language is being used. 'The total sum' is not the same thing as 'equals'.

The total sum cannot be explicitly demonstrated. It is really a concept that cannot be substituted with 'equals'.

the ... means the limit is taken. IE, .999... = 1 — fdrake

Yes but 'limit' is not the same as 'equals'.

It comes down to how the language is used .9999... equals 1. This is not true. In calculus one is only allowed to say that a series converges to a limit. In this case the series is .999... and the limit is 1. That is, incorrectly speaking, .999...becomes one after an infinity of 9s. But you can't have 'an infinity of' 9s because infinity is not a number. So no, .999... is not 1. It only converges to 1. That is all the link you posted is showing. They are replacing 'tends towards the limit 1' with 'equals'.
You can say The limit = 1 but not .999...= 1

Thought is being and being is thought. In our modern ossified world we tend to believe thinking is just abstract machinations and 'logic' and such things. It is much broader than that.

Science has shown remarkable capability of verification, prediction and use.
How is is this possible if it is only the appearance of external reality (phenomena), not the external reality itself (noumena)? — Arthur Rupel

Look at it this way. If we surmise that what we perceive is analogous to what is really out there we may be able to make progress. For example, if we see something moving in a circular manner we can surmise that there is something out there that is changing in a cyclical manner - not necessarily a circular manner. So our experience is telling us something real.

Also, if we perceive something, assuming the analogy as above, we can then focus on a smaller detail of the perceived object. Then on a smaller detail and so on. We can then determine how these details are related to each other and assume that these perceived relationships between parts are also analogous to the actual relationships between the parts of the perceived object.

This process of endless division into smaller parts should be coherent and analogous to the coherence between the parts of the object itself. If there is coherence in our phenomenological perception, down to the finest detail, this coherence can be understood to reflect the actual coherence of the object itself because the object is the source of our perception.

In other words, if our perception is, in any way, analogous to what is really there, it should be analogous down to smaller and smaller details. We can then make a detailed map of what is really there.

The only other possibility is that nothing in our perception is analogous to what is actually there, which is philosophy gone mad.

So it doesn't mean anything? — tim wood

Isn't child abuse perverse? I don't see why one's subjective point of view needs to be meaningless. The sadist is acting out evil. The masochist wants freedom from self - without giving up self.

What is? — tim wood

That depends on one's subjective point of view.

if I adduce enough arguments to show that time is unreal, time might stop. In other words, there is a recognition that since one can speak however one pleases, that one can in some sense 'make true' whatever one pleases, just by talking about it. — Snakes Alive

Isn't this exactly what Lazerowitz is doing? Talking himself into his own truth? Maybe all philosophers do that, lol.

Sexuality is spirituality in bodily terms. Sexual distortion is spiritual distortion. Think about it...

What I've referred to as the (mono)theistic 'command to love' seems akin to masochistic rape-fantasy or self-abnegation: — 180 Proof

Masochism is a distortion of spirituality, not the other way round.

You have a point there I suppose. What means you by wise obedience? — TheMadFool

It means thinking about why religion tells us how to live. This is what Buddhism means by 'wise living'. But, of course, sometimes religion is corrupt.

To entertain the idea of false/misguided obedience is to sow the seeds of disobedience that ultimately leads to the rejection of god. — TheMadFool

The hope is that it would lead to wise obedience. Slavish obedience is not a good thing. Obedience should be understood rather than rejected at face value. There is a worthy goal in wise obedience.

one of the most effective methods to make normal people do immoral things is to convince him/her that s/he is doing god's will. — TheMadFool

That would be false or misguided obedience.

Are you defining discipline as obedience to one's self? — Pantagruel

Yes, but obedience to someone else can also be a good thing if that obedience has a purpose. An athlete needs to listen to and obey the trainer. Musicians need to be attentive to the conductor. Pupils needs to be obedient to the teacher. But obedience should be understood and its purpose needs to be understood. In evil obedience is slavery to a tyrant. But there are times when obedience is wise and good.

Isn't this slavery? A slave must obey his master's command and the master makes it clear that he has zero tolerance for any disobedience. — TheMadFool

It depends on what you mean by obedience. Obedience is essential for development. eg someone practicing music or art must have great discipline/obedience. A great tennis player must make their body obedient to their mind.
Wise obedience is a good thing if it leads to freedom. A good soldier must learn obedience. A bad soldier will get killed. One must try to understand why obedience is necessary and how it leads to freedom. Sloth and disobedience don't achieve anything worth while.

The vast majority of reality at every scale is space, that which we typically call nothing, or non-being. — Nuke

In our simple human definition of being, yes. God is energy and being. Space is God. There is nowhere where God is not (space is no-thing but not nothingness (there's a difference) It has positive being.)

Every act of creation is an act of destruction, and every act of destruction is an act of creation. — Nuke

Ultimately everything is evolving and moving forward. But you cannot judge these things in human terms. Even good and evil become unclear in the hurly-burly of human affairs. Ultimately being and life emerge from matter, time and space. We'll have to wait till the end to see how it all works out.

That was very poetic. I like your personal take on it. — Benj96

Glad you enjoyed it.

Wouldn't good have to be in reference to something? You know, good for who or what? — Nuke

The existence of the good as evidence for God's existence is hard to argue with. But yes, what is the good? Ultimately the good promotes life and being. Evil is ultimately non being. The good is unity. Evil is egoism and separation; the black hole detaches itself from everything else and becomes darkness.

Then how did energy ever give rise to mass (e=mc2)? If it cannot do anything to itself in a state of pure timelessness then how did it just spontaneously slow down and get "heavy" with matter in the first place. — Benj96

That's Higgs theory; mass is a slowing down of energy. Mass and matter are not substances, they are processes.

$\exists$ x = $\frac{1}{a_1^\frac{p - 1}{2}(mod p)}$
such that x = $\pm 1$

${\sqrt {ab \pm c} \over \sqrt{x/y}} \pm 1$

$a \equiv b (mod m)$

$\phi =$$\frac {1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}$

$\sum_{i=1}^{\infty}$

${\sqrt {2 - 1/\sqrt {2}}}$

2. The magnitude of the effect is proportional to the magnitude of the *effect* — TheMadFool

Shouldn't that be 'cause'?
Corruption moves away from the good which is love, life and being. Corruption moves towards hatred, death and non being. This is despair, hell. Call this karma if you will...

You sound like the opposite to a misanthrope; you believe in humanity and the fact that you see how people's behaviour is wrong shows you believe in standards of behaviour. As for being irritated - there will always be such people. Personally I choose to ignore them as far as possible. Gravitate towards better folk.

Thus, consciousness is needed to make physical reality meaningful.

Well, that goes without saying. How can meaning exist outside a mind? I think he is talking about something slightly different here.

'What did you do to the cat, Erwin? — Wayfarer

Schrodinger's Cat was not originally meant to be taken seriously. It was an illustration of the problems with emerging quantum mechanics at the time.

A lot of this confusion about things not happening until we 'observe' them has to do with a conflation between detection and observation, as if they were the same thing. But they are not.

Observation is when a scientist looks at the results of an experiment. That has nothing to do with how the experiments takes place or what the results are.

Detection is when a particle in the quantum universe/spacetime collides with an object (experimental apparatus) in physical spacetime. This is what determines the result of the experiment. Observation has nothing to do with it. After all, a scientist could wait months before observing the results of an experiment. Does that mean there are no results until he observes those results? I don't think so.

A lot of pseudo books have been sold on this 'changing reality by observing it' notion...

Patterns in mathematics are often simplifications. Also, some patterns apply in infinitely many situations. For example, if p is prime, a^p leaves a remainder a when divided by p: 3⁵ = 243 which leaves remainder 3 when divided by five.

This applies to all primes so we can say 733⁶¹⁰⁶⁹⁹ leaves remainder 733 when divided by 610699. We can know this without doing the calculation. Simplicity and universality of patterns in mathematics are what the mathematician seeks.

The category is bolded. See left. — Outlander

Ah. Thank you.

At least in the android store. — Benkei

Thanks.

check out the app xsection — Benkei

Where is that?

It is possible for both to suspend empathy. In fact, the psychopath is not devoid of empathy (that's almost an urban myth). The psychopath has repressed empathy and conscience with such ferocity he does not feel it. He supresses it because if he didn't he could not indulge in his monstrous behaviours.

If you commit a crime against a psychopath s/he will tell you in no uncertain terms what you did was wrong and why it was wrong and how you should be punished. But s/he forgets all this when it is the other way around.

imagine six crosses arranged in two rows of three crosses each, one row directly above the other. I can equally imagine the same six crosses as three columns of two each. Therefore 2 × 3 = 3 × 2. I not only notice that 2 × 3 is in fact equal to 3 × 2, I understand why 2 × 3 must equal 3 × 2. — The mathematical world - James Franklin

This is an example of induction (observation) awakening our powers of deduction. Resulting in deducing the commutative nature of multiplication: 2 x 3 = 3 x 2.

This visual proof is a bit more elegant: — InPitzotl

Very nice. I had not seen that one before.

That's actually a beautiful picture of things. I think it leads to a problem of evil though. Do you have a solution? — frank

Thank you. Because good (being) exists, distortions of good exist. As St. Augustine said, evil is not a positive thing in itself. It is a lack of the good. The good is a perfect symphony. Evil is disharmony. But evil cannot exist without the good, without being, which is God.