, I just gave a proof involving a sequence that gives the exact value of the limit: zero. $\lim_{n \to \infty} \frac{1}{n} = 0$

This is a counter instance to your insistence. The "=" is not an approximation.

So if you would keep your credibility, show your working.

Damn keyboard keeps sticking.

Then why did you say to@jgill, "a more intricate form of 'rounding off'"? — Metaphysician Undercover

Because he was looking at Numerical Analysis not Real Analysis.

Ok. Details?

Simple example of a limit with an exact value
Consider the sequence
$a_n = \frac{1}{n}$

Claim
$\lim_{n \to \infty} \frac{1}{n} = 0$

Proof (ε–N)
Let $\varepsilon \gt 0$ be arbitrary.

Choose $N \gt \frac{1}{\varepsilon}$.

Then for all $n \ge N$,
$\left| \frac{1}{n} - 0 \right| = \frac{1}{n} \lt \varepsilon$

Since this holds for every positive $\varepsilon$, the difference between $\frac{1}{n}$ and $0$ can be made smaller than any positive real number.

Therefore,
$\lim_{n \to \infty} \frac{1}{n} = 0$

Conclusion
There is no “infinitely small but non-zero” remainder. In ℝ, being smaller than every positive real number forces equality with zero.

With the help of ChatGPT. Let me know if it's wrong. Looks OK to me.

Well, you can play with all that if you like - some of what you say here looks muddled. The salient bit today is that a limit is not a rounding off.

To which we might add, as a corollary, The limit is not “almost” the value.

well I haven’t had an exam on it in 50 years…

Not just Cauchy.


Tell me where I’m wrong if you can.

And the answer is that one sees the apple by constructing a representation of the apple. — Banno

There's a need to be clear here that representation is Michael's word. Neural nets of course do not function by representing one thing as another. they function by modifying weightings. It’s just a pattern of activations and weights, with no intrinsic “aboutness” or semantic content.

Better to say they model, in a statistical, functional sense.

:smile:

Back a few pages I began a bit on the definition of a limit. I got as far as completeness and the least upper bound. Every nonempty set of real numbers that is bounded above has a least upper bound in ℝ, the smallest real number that is greater than or equal to every element of the set. It's the existence of this number that guarantees the existence of a limit when one uses the sequence in a calculation... if it's a monotone increasing sequence that is bounded above...

But as you found, the interesting stuff is the variations on these themes. The thread is focused on a small, very specific region of maths, and mostly failing to get a good handle on even that.

Interesting. A worthy topic - a more intricate form of "rounding off"? :wink:

I'll defer to your experience. My understanding is that what I said holds for classical convergence in Real Analysis.

Given that "I see X" is true if "I indirectly see X" is true, it is a non sequitur to argue that if "I see X" is true then "I directly see X" is true. — Michael

But the argument is not that I directly see X, because that is little more than a rhetorical ploy on the part of the indirect realist. At issue is whether one sees the apple or a representation of the apple.

And the answer is that one sees the apple by constructing a representation of the apple.

I could say "I saw Alcaraz defeat Djokovic in tennis" or I could say "I saw images on my computer screen". — Michael

Yep. Different placements of the Markov Blanket.

What we should not say is that we never saw Alcaraz defeat Djokovic, only ever images of Alcaraz defeating Djokovic.

It's an example of seeing an apple without an apple being a constituent of the experience. — Michael

:meh: This gaslights itself.

In your example, the apple causes the pattern of light that is seen ten seconds later. Hence the apple is a constituent of the experience.

No they don't. — Michael

So "I see X" is true if we directly see X or if we indirectly see X and yet they do not collapse into one? Not following that at all.

So you say "I see the apple" is true, and so is "I see the mental representation of the apple", and you want to claim these are the same? But it is clear that an apple is different to a mental representation of an apple. You can't make a pie with a mental representation.

Going over the already dispelled though experiment doesn't help you here.

No I don't. "I see X" is true if we directly see X or if we indirectly see X. — Michael

Good. then the two collapse into one. And you have now agreed that "I see the apple" is true, and "I see a mental image of the apple" misleading. "first-person phenomenal experience" is philosophical fluff.

Naive realists say that apples are "constituents" of first-person phenomenal experience... — Michael

So indirect realists say that apples are not "constituents" of our seeing apples? How's that?

So the strawberry is actually grey?

Notice how you here work with the merely philosophical construct "the-strawberry-as-it-is-in-itself"? We never get to taste or see "the-strawberry-as-it-is-in-itself" not becasue of any limitation on our senses, but becasue it's not a thing. "the-strawberry-as-it-is-in-itself" is already interpreted.

As being grey, apparently.

You always conflate "I see an apple" and "I directly see an apple". — Michael

Hokum. You conflate "I see an apple" and "I indirectly see an apple".

Again, that "naive realist" is no more than a foil against which to draw the supposed "indirect" account. That indirect account is misleading. What one sees is the apple, not a mental image or whatever.

No it isn't. Indirect realism says that what we see is not the apple.

It suffices as a refutation of indirect realism. What we see are apples, not mental images of apples. Seeing an apple is constructing a model of that apple. That model is of the apple, and is not what is seen.

The representation is the machinery, not the seen object.

At 10:00:25 there is no apple, only first-person phenomenal experience with subjective character — described as "seeing a red apple" — and this first-person phenomenal experience with subjective character is a mental representation of an apple that no longer exists. — Michael

Notice that the conclusion, that we see "only first-person phenomenal experience with subjective character", is not argued for but merely asserted? You are repeatedly presuming that what we see is a "first-person phenomenal experience with subjective character", and not an apple.

That's not how language works. We can talk about the first-person phenomenal experience, but that does not mean that we cannot talk about the apple, including seeing the apple.

Your example continues to confuse the causal chain with the epistemic outcome.

You are misrepresenting the grammar of "seeing a mental representation". — Michael

Seeing an apple is constructing a mental representation, if you like - it depends where one places the Markov Blanket. But one does not see a mental representation. One sees an apple.

light doesn't move — DifferentiatingEgg

:meh:

What does it mean to see the apple as it was? — Michael

Just that.

Again, what we see is the apple, and not a memory, a sense datum, a representation, an image,
or anything “in the mind”. Sure, the causal chain that is seeing the apple includes a delay, but so what.

Your argument is merely rhetorical, a play on the word "direct". What one sees is the apple with a ten-second delay. What one does not see is some mental representation of the apple as it was ten seconds ago.

The difference between the limit and the sum is an infinitely small number. — frank

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.

The latter can be understood as "rounding off". — Metaphysician Undercover

No.

There is no principled theory/practice gap here. “Approaches the limit” and “equals the limit” are not in tension. Introducing “rounding off” does not correct or deepen the mathematics—it changes the subject.

The infinite sum of the geometric series (1,0.5,0.25,...) is technically undefined, for in this case, every partial sum S(n) is non zero, since S(n) = 2 - 0.5^(n-1). — sime

The fact that no partial sum equals 0 does not imply anything about whether the limit exists, or what it is. Limits routinely exist even when no term (or partial sum) ever equals the limiting value.


  The infinite sum of a series is defined as the limit of its partial sums (when that limit exists):

$\sum_{k=1}^{\infty} a_k := \lim_{n \to \infty} S_n$

For the geometric series

$1 + \tfrac12 + \tfrac14 + \cdots$

the partial sums are

$S_n = 2 - \left(\tfrac12\right)^{n-1}.$

Since

$\left(\tfrac12\right)^{n-1} \to 0 \quad \text{as } n \to \infty,$

it follows that

$\lim_{n \to \infty} S_n = 2.$

The fact that every partial sum is non-zero is irrelevant; convergence depends on the existence of the limit, not on whether any partial sum equals the limiting value.

Even on this “game” interpretation, the geometric series trivially has a winning strategy for every
ε. So by your own account, the limit exists.

The argument with the slow light is merely to show that (1) is false — Michael

If you include the assumption that direct perception requires temporal coincidence between perceiver and perceived. There is no need to do so.

What one does not see is some mental representation of the apple as it was ten seconds ago.

But even to entertain that scenario is a step too far. Temporal lag does not introduce a new object of perception. We see the apple as it was, and not a memory, a sense datum, a representation, an image,
or anything “in the mind”.

You mean the key is to put an end to the infinite sequence by rounding off. — Metaphysician Undercover

No. Nothing to do with rounding.

Your failure to understand mathematics is not our problem.

I'm not seeing why this should be difficult. The suggestion was that someone traveling at near-light speed would have a different experience to someone at rest. Pretty clear that's a violation of the Principle of Relativity.

The proof: you are traveling at near light speed relative to some frame of reference, yet you do not experience any difficulty.

End of thread.

It comes back to whether what we see is an apple or a mental-image-of-an-apple. But that's a infelicitous question.

:wink:

It would appear slowed down.

That's what he equations say, and what empirical observation supports.

Their physical processes cannot be observed from my frame until they return. — Janus

Watch them on TV.

fucksake.

What, if anything, in the supposed paradoxes of motions from Zeno, is not answered by limits, infinitesimals and calculus?

I suggest that what does remain is not a problem about motion, space, or time, but about conceptual confusion over infinity, divisibility, and description.

The key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1

Velocity is not defined at an instant by a finite displacement, but as: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

The theory says that if you traveled at the speed of light to some distant star and then returned, those on Earth would have aged much more than you. In that scenario Earth is the stationary, "normal" frame and the starship the one at great speed relative to it. — Janus

The twin paradox is a result of the relative acceleration of the traveler. The OP is only asking about inertial frames of reference. You are adding an unhelpful complication.

Consider:

Suppose I could somehow observe their inner mental activity directly. — RogueAI

Such an observation would be mediated by a signal from observed to observer. That signal is either subject to the Lorentz transformation, in which case the time dilation takes effect, or it isn't, in which case there is an absolute frame of reference.

So 's hypothesis does assume an absolute reference frame in presuming frame-independent access to temporal structure. It adopts a privileged temporal standpoint.

Michael has used a bit of rhetoric to put those opposed to indirect perception on the back foot. They feel obliged to defend "direct" realism.

What one sees is the apple with a ten-second delay. What one does not see is some mental representation of the apple as it was ten seconds ago.

If someone were traveling close to the speed of light relative to me... — RogueAI

From their frame of reference it's you who is traveling close to the speed of light. Are your thought processes slowed in respect to the movement of your body?

What your thought experiment shows is a misunderstanding of the issue. You continue to suppose some frame of reference that is stationary in an absolute sense.

Falsification was first developed by Karl Popper in the 1930s. Popper noticed that two types of statements are of particular value to scientists. The first are statements of observations, such as 'this is a white swan'. Logicians call these statements singular existential statements, since they assert the existence of some particular thing. They can be parsed in the form: there is an x which is a swan and is white.

The second type of statement of interest to scientists categorizes all instances of something, for example 'all swans are white'. Logicians call these statements universal. They are usually parsed in the form for all x, if x is a swan then x is white.

Scientific laws are commonly supposed to be of this form. Perhaps the most difficult question in the methodology of science is: how does one move from observations to laws? How can one validly infer a universal statement from any number of existential statements?

Inductivist methodology supposed that one can somehow move from a series of singular existential statements to a universal statement. That is, that one can move from ‘this is a white swan', “that is a white swan”, and so on, to a universal statement such as 'all swans are white'. This method is clearly logically invalid, since it is always possible that there may be a non-white swan that has somehow avoided observation. Yet some philosophers of science claim that science is based on such an inductive method.

Popper held that science could not be grounded on such an invalid inference. He proposed falsification as a solution to the problem of induction. Popper noticed that although a singular existential statement such as 'there is a white swan' cannot be used to affirm a universal statement, it can be used to show that one is false: the singular existential statement 'there is a black swan' serves to show that the universal statement 'all swans are white' is false, by modus tollens. 'There is a black swan' implies 'there is a non-white swan' which in turn implies 'there is something which is a swan and which is not white'.

Although the logic of naïve falsification is valid, it is rather limited. Popper drew attention to these limitations in The Logic of Scientific Discovery, in response to anticipated criticism from Duhem and Carnap. W. V. Quine is also well-known for his observation in his influential essay, "Two Dogmas of Empiricism" (which is reprinted in From a Logical Point of View), that nearly any statement can be made to fit with the data, so long as one makes the requisite "compensatory adjustments." In order to falsify a universal, one must find a true falsifying singular statement. But Popper pointed out that it is always possible to change the universal statement or the existential statement so that falsification does not occur. On hearing that a black swan has been observed in Australia, one might introduce ad hoc hypothesis, 'all swans are white except those found in Australia'; or one might adopt a skeptical attitude towards the observer, 'Australian ornithologists are incompetent'. As Popper put it, a decision is required on the part of the scientist to accept or reject the statements that go to make up a theory or that might falsify it. At some point, the weight of the ad hoc hypotheses and disregarded falsifying observations will become so great that it becomes unreasonable to support the theory any longer, and a decision will be made to reject it.

In place of naïve falsification, Popper envisioned science as evolving by the successive rejection of falsified theories,rather than falsified statements. Falsified theories are replaced by theories of greater explanatory power. Aristotelian mechanics explained observations of objects in everyday situations, but was falsified by Galileo’s experiments, and replaced by Newtonian mechanics. Newtonian mechanics extended the reach of the theory to the movement of the planets and the mechanics of gasses, but in its turn was falsified by the Michelson-Morley experiment and replaced by special relativity. At each stage, a new theory was accepted that had greater explanatory power, and as a result provided greater opportunity for its own falsification.

Naïve falsificationism is an unsuccessful attempt to proscribe a rationally unavoidable method for science. Falsificationism proper on the other hand is a prescription of a way in which scientists ought to behave as a matter of choice. Both can be seen as attempts to show that science has a special status because of the method that it employs. — Banno

Summary: falsification applies to universal sentences, such as ∀(x)(fx⊃gx). For an open domain, no amount for evidence can show that everything that is f is also g. But a single example of an f that is not g can serve to falsify it. Hence, falsificaton serves to show which universals are false, but not which are true.

Universal sentences cannot be shown to be true in an open domain. If the universal sentence is also for some reason unfalsifiable then it cannot be shown to be either true or false.

Caveats:

• The claim only works if restricted to synthetic empirical universals in open domains.
• Shown to be true/false here means being empirically demonstrated.

This is basic, naive fallibilism at it's core. All sorts of complications and complexities follow. But understanding the logic of falsification is central to following argument concerning scientific method.

Travesty.

:grin:

No one will get that reference.

Cheers.

Does anyone care whether the right whingers vote for The National Party or One Nation?

And this. Hanson has made herself a movie.

:meh:

@Jamal, so how goes it?