Actually, I was trying to understand it myself. It's tough finding an explanation in plain english.

Fuck off. — Banno

Oh dear. It seems you can't.

Would it be better to attack it without understanding it? — Banno

No. Can you explain the difference between point-wise and uniform convergence?

You're aware that the issues of the century before last were solved using an axiomatisation of the continuum - along the lines started earlier in this thread - and then nonstandard analysis showed they weren't such a problem, anyway...?

So...? — Banno

I read about it, yes. Cauchy's original solution was eventually rejected in favor of Weierstrass's solution. As has been mentioned, this is history that usually isn't taught. I know this irritates you, but what's most interesting to me is the way people defend it when they don't actually understand it.

If representation is just the function of neural states, your philosophical view stops being indirect and becomes more of scientific direct realism. — Richard B

No.

And you suppose that to be an end to it? — Banno

Absolutely. Never to come up again. :grin:

My first calculus teacher was awesome. He told us this story about when he was young and he dropped some mercury on the floor. They tried to sweep it up with a broom and it turned into a "blue bloody million" little balls of mercury.

The history of mathematics is a worthy subject in itself, but that is not the topic here. — SophistiCat

Cauchy, Weierstrass, and Riemann saved calculus from mounting criticism that it doesn't make sense. So, that did become the topic. We've finished talking about it now, though. :grin:

The limit will be called the sum of the series. — Banno

Yes. I don't think either of us is interested in arguing about what that actually means. Let's leave it.

But the point I made is that "point particle" is a conception of convenience, designed for the purpose of representing interactions. It does not represent how the electron is actually conceived as existing. The electron is modeled as a "point particle", but it does not exist that way, the probability cloud is a better representation (though still very inadequate) of how electrons exist. — Metaphysician Undercover

I actually took a deep dive on this at one point. The electron is, in fact, conceived by scientists as a point. It's startling, but true.

You're right that per Cauchy, the sum of the series is the limit. However, the devil is in the details. The sum and the limit are never equal. see here.

I'm not concerned about credibility or showing that I'm working. :grin:

But an electron is conceived as a point.
— frank

I don't think so — Metaphysician Undercover

It is. An electron is a point particle.

Ok. Details? — Banno

Yea, you're wrong.

It's also an answer to this, I think. — Srap Tasmaner

How so?

The limit is not “almost” the value. — Banno

That's incorrect.

Tell me where I’m wrong if you can. — Banno

According to Zvi Rosen, the sum and the limit are not equal (according to Cauchy). They're just as close as we "want" them to be. This is Cauchy's definition of a limit:

When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others. — Cauchy

So it's true that the idea of an infinitesimal was removed, but the idea of infinitely small remained, and we added the idea of "as small as we want."

We'll see what the other's said.

Wait, what? — AmadeusD

You don't have confidence that you can tell what's true and real?

That is to say, none of this discussion is responsive to the metaphysical question of what the fundamental constitution of reality is. As in, what is the apple in the noumena? — Hanover

That is correct. Both sides of this argument start with irrational confidence in our ability to discern what is true and real. Neither side proposes to build a bridge to that confidence. As you noted, there is no bridge to it. You just have it.

Starting with that confidence, we observe by way of anatomy and physiology that perception of the world appears to be constructed by the brain out of discreet electrical impulses. As you note, this is not a metaphysical argument, it's a scientific fact.

For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit. The sum is the limit. Partial sums are less than the limit, but their difference goes to zero in the standard real number system. — Banno

This is all from proofs by Cauchy that I don't understand. Do you understand it?

For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit — Banno

Ok.

I don't agree. I think the average scientist would say that it doesn't make sense to talk about infinitely short distances — Metaphysician Undercover

But an electron is conceived as a point. It doesn't have any length or height. Isn't that the same as the idea of an infinitesimal in math?

It is a difference between theory and practise. In theory, the sum approaches the limit. In practice the sum is the limit. The latter can be understood as "rounding off". Failure to recognize this is to misunderstand. — Metaphysician Undercover

It's not an ordinary sort of rounding off, though. The difference between the limit and the sum is an infinitely small number. We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle.

I'd say I directly perceive pain, colours, smells, tastes, etc. — Michael

The experience of pain is generated the same way the experience of seeing an apple is.

I didn't say that. — javi2541997

That's cool. So you know you aren't better anyone else. Anywhere. Ever.

I don't think there's any such thing as direct perception. The only perception there is is indirect.

We are not perfect, but at least we are not like you. — javi2541997

Simmer down there Franco, you aren't better than anyone else.

I mean lay out the methodology of this experiment, show me what we're measuring, and show me the results we have to arrive at to prove direct realism is true.

My point just being that the question is nonsense. It can't be proved in principle. It's unverifiable, just as is indirect realism is unfalsifiable. You would have to assume indirect realism to even perform an empirical analysis, considering empirical measurement relies upon perception.

For some reason this thread conflates "physical" with "metaphysical." Telling me we describe apples in the physical world as X doesn't tell me the fundamental nature of things. It can't. — Hanover

Some people wouldn't be able to accept what you're saying because it would mean they have to accept that the world may be a dream. We really don't know. It's emotion that makes them cling fiercely to a particular worldview, so they don't care if what they're saying is nonsense.

I think there's an incomprehensibility field around Wittgenstein's Tractatus. Even after a person understands it, they'll say yes, but.. the world is this or that. Emotion drags the mind around. The only way to free the mind is to admit: I don't know.

:grin:

What experiment would prove the validity of direct realism as you define direct realism? — Hanover

It would have to show that the world is actually a dream. You directly perceive the world because there's no interface. The world is created by you, from the contents of you. It would have to show that you're God.

BaShana, BaShana, BaShana

I think that this "Fort Sumter"- — ssu

A Fort Sumter moment would require that half the states had already seceded. :cool:

No, you described a long process, and the problem is with the use of "at some point". How does a process occur at a point? — Metaphysician Undercover

I think you just need some more of whatever mind altering substance you have available. Then you'll get it.

Since the Ukraine war started, Russia has engaged in "systematic dismantling of independent civil society, harsh suppression of anti-war dissent, and the persecution of minority groups."

In the US, we're about to explode because 2 people were murdered by ICE. That's actually a good sign about the health of rule of law.

would say the opposite is the case, what you say sounds bizarre. You are representing driving through British Columbia, as being in British Columbia at some point. What does "at some point" even mean in this context? You use it because it's an acceptable figure of speech, but taken literally, it doesn't fit. So what does it really mean? — Metaphysician Undercover

Being in British Columbia usually entails waking up in your car with a Canadian citizen tapping on your window to see if you're ok. You roll your window down and try to do a Canadian accent so they don't know you're American, at which point they just stare at you. Does that explain it?

Yes and this is, as you probably know, one of Nietzsches main issues with a purely utilitarian view on morality. We need some adversity to be able to grow. The quest to reduce all suffering would ultimately also reduce what we can be as human beings. — ChatteringMonkey

:up:

This is the real natu — Metaphysician Undercover

Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre.

I read you as basically saying there's a higher truth missed by Zeno. Per tradition, his point was exactly that: that the way we picture the world, the way we commonly think, is missing something.

If you go back and look at one of the paradoxes, they're pretty simple. Looking at in terms of truth it starts here:

Isn't it true that in order to get from point A to point B, you have to travel half the distance between them?

Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? If you say no to that, you've already ejected yourself from common sense. If you say yes to it, you're on your way to being ejected from common sense because there's a convergent series of points between. Either way: common sense has a problem.

There are two kinds of people: ones who can tolerate a threat to common sense, and those who can't. I think the first category is usually non-linear thinkers.

A Euglena is a one celled thing that has both mitochondria and chlorophyll. It's hard to kill because if it has light, it gets energy that way. If there's no light, it can eat.

Plants and animals both went through all sorts of evolution trying to survive, and in the process, became complex, but they're still precarious compared to the Euglena (algae).

Which just goes to show: we humans exist because of adversity. If we had utopia, we would just sit there.

Supertask. It's the reason Zeno's paradox stands.

:up:

I don't think anyone in this thread had forgotten, or that anyone was confused. — Srap Tasmaner

I think I could find cases of it in this thread. I'm not going to mine it to find them though

Some people reject talking about infinite collections — Srap Tasmaner

And since you bring that up, let's look at the difference between a collection, an extensional definition, and a set. Just because I think we need to stuff that difference down this thread's throat. :blush:

The extension of an idea need not be thought of as an abstract object. A set has to be thought of that way. There's no choice. The people who invented set theory knew that.

Which some authors prefer, but it means what other authors mean by "countable". So long as we know what we mean, "The natural numbers are violet" would do just fine. — Srap Tasmaner

Absolutely. Let's keep in mind that it does not mean the same thing as countable as the word is commonly understood.