Yeah, it might have helped to clarify that the infinitesimal factors I had in mind are not infinitely small as numbers from the standpoint of addition. Since the factor that makes no change to the product is 1 rather than 0, "infinitely small" factors must be infinitesimally greater than 1, not 0. In particular, I was talking about a Type II product integral with the formula pi(1 + f(x).dx). If f(x) = 1, then we get e^sigma(1.dx) = e^constant = constant, right?
No, he's right. I didn't think to clarify that my infinitely small factors are infinitesimally larger than 1, not 0. See the Type II product integral formula on Wikipedia that uses 1 + f(x).dx.
Thanks, product integral is what I was talking about. The exponentiated integral is what I meant when I said the integration will move into the power term.
Someone has probably thought of this already, but if we defined an integration analogue where larger and larger logarithmic sums cause their exponentiated, etc. value to approach 1 rather than infinity, then we could use it to define a really cool account of logical metaphysics: Each possible state of affairs has an infinitesimal probability, there are infinitely many of them, and their probabilities sum to 1. This probably won't be exhaustive in some absolute sense, since no formal system is both consistent and complete, but if we define states of affairs as formulas in some consistent language, then why not? We can then assign various differential formulas to different classes of states of affairs.
(That is the context in which this came up. The specific situation is more technically convoluted.)
Integrals sum over infinitely small values. Is it possible to multiply infinitely small factors? For example, Integration of some random dx is a constant, since infinitely many infinitely small values can sum up to any constant. But can you do something along the lines of taking an infinitely large root of a constant, and get an infinitesimal differential in that way? Multiplying those differentials will yield some constant again.
My off the cuff impression is that this probably won't lead to genuinely new math. In the most basic case, all it does is move the integrations into the powers that other stuff is raised by. But if we somehow end up with complicated patterns of logarithms and exponentiations, like if that other stuff itself involves calculus and so on, then who knows? Is there a standard name for this operation?
I don't see how you can achieve a reductionist ontology without positing a hierarchy of qualities. In order to propose a scientific reduction, we need at least two classes, one of which is reducible to the other. Perhaps "physical" and "perceived" qualities would be more specific than "primary" and "secondary" qualities.
Regarding your question, if the "1->2 and 1->3" theory is accurate, then I suppose when we say that "red is more like violet than green", certain wavelength ranges R are causing the human cognitive architecture to undertake some brain activity B that drives both the perception of color similarity A a well as behavior which accords with perception C.
So it follows that... (read more)
I don't think epiphenomenalists are using words like "experience" in accordance with your definition. I'm no expert on epiphenomenalism, but they seem to be using subjective experience to refer to perception. Perception is distinct from external causes because we directly perceive only secondary qualities like colors and flavors rather than primary qualities like wavelengths and chemical compositions.
EY's point is that we behave as if we have seen the color red. So we have: 1. physical qualities, 2. perceived qualities, and 3. actions that accord with perception. To steelman epiphenomenalism, instead of 1 -> 2 -> 3, are other causal diagrams not possible, such as 1 -> 2 and 1 -> 3, mediated... (read more)
In the Less Wrong Sequences, Eliezer Yudkowsky argues against epiphenomenalism on the following basis: He says that in epiphenomenalism, the experience of seeing the color red fails to be a causal factor in our behavior that is consistent with us having seen the color red. However, it occurs to me that there could be an alternative explanation for that outcome. It could be that the human cognitive architecture is set up in such a way that light in the wavelength range we are culturally trained to recognize as red causes both the experience of seeing the color as well as actions consistent with seeing it. After the research which shows that we decide to act before becoming conscious of our decision, such a setup would not be a surprise to me if true.
Thanks. You're right, that part should be expanded. How about:
At this point, you have two choices: Either 1. one randomly selected door, or 2. one door among two doors, chosen by the host on the basis of the other not having the prize.
You would have better luck with option 2 because choosing that door is as good as opening two randomly selected doors. That is twice as good as opening one randomly selected door as in option 1.
(Disclaimer: I have no training in or detailed understanding of these subjects. I first heard of Tarski from the Litany of Tarski, and then I Googled him.)
In his paper The Semantic Conception of Truth, Tarski says that he analyzes the claim, '"Snow is white" is true if and only if snow is white' as being expressed in two different languages. The whole claim in single quotes is expressed in a metalanguage, while "snow is white" is in another language.
For Tarski's proof to succeed, it is (if I understood him correctly) both necessary and sufficient for the metalanguage to be logically richer than the other language in certain ways. What these ways are... (read more)
I'm not sure that analogy can be extended to our cognitive processes, since we know for a fact that: 1. We talk about many things, such as free will, whose existence is controversial at best, and 2. Most of the processes causally leading to verbal expression are preconscious. There is no physical cause preventing us from talking about perceptions that our verbal mechanisms don't have direct causal access to for reasons that are similar to the reasons that we talk about free will.
Why must A cause C for C to be able to accurately refer to A? Correlation through indirect causation could be good enough for everyday purposes. I mean, you may think the coincidence is too perfect that we usually happen to experience whatever it is we talk about, but is it true that we can always talk about whatever we experience? (This is an informal argument at best, but I'm hoping it will contradict one of your preconceptions.)