KJW

This is drawing a longbow. Choosing examples where cognition is important to survival isn't sufficient to show that cognitive abilities are essential in natural selection. What about examples where cognition plays no role in survival? For example, what cognitive abilities do trees have? And yet trees are also evolutionarily successful. The various ways that organisms can become evolutionarily successful is unlimited. For example, animals such as cattle, sheep, pigs, and chicken have become evolutionarily very successful simply by being good food for humans. I doubt that is a choice those animals would have made cognitively.

Hmmm. I did not realise THAT was what I had proven. From what I said above, proving that a map from G to {1, –1} is a homomorphism completes the proof that a simple group cannot have subgroups of order half that of the group.

This actually seems like a bad idea to me.

I can't speak about all groups G, but if there is a homomorphism that maps G to {1, –1}, then the kernel of the homomorphism (the subgroup of G that maps to 1) is a normal subgroup of G that contains exactly half the elements of G. In particular, the alternating group An of all even permutations of n objects, is a normal subgroup of the symmetric group Sn of all permutations of n objects. It should be noted that Cayley's theorem states that every group is isomorphic to a subgroup of a symmetric group. It occurred to me that the alternating group An where n is greater than or equal to 5 is a simple group of even order. That is, it has no normal subgroups other than the trivial group and the group itself. Thus, if the alternating group has any subgroups of order half that of the group, then the answer to your question is no. But there might not be any subgroups of order half that of the group.

And now that I'm an adult, I wonder why couples in TV shows always cover themselves when getting out of bed after having sex.

I chose not to look at the linked video, but I was curious, nevertheless.

What is the purpose of this thread? When I saw first saw it, I thought it might have been about censorship, or perhaps the notion of privileged knowledge. But then I thought it might be about preserving treasures. For example, an art museum is not going to loan out the Mona Lisa to the general public.

Then k and n are relatively prime.

No, the group generated by another element is a subgroup of G, not necessarily G itself. k is a factor of n (or perhaps a multiple of a factor of n).

The importance of normal subgroups is the following (which I won't explain at this time): Let G be a group and N be a normal subgroup of G. Then there exists a quotient group or factor group G/N and a homomorphism that maps G to G/N, and under this homomorphism, N maps to the identity of G/N. Conversely, for every homomorphism of group G, there exists a normal subgroup N such that N maps to the identity of the image of the homomorphism, and a quotient group or factor group G/N that is isomorphic to the image of the homomorphism. g1 = h1 g h1–1 g2 = h2 g h2–1 h2–1 g2 h2 = g g1 = h1 g h1–1 = h1 h2–1 g2 h2 h1–1 = (h1 h2–1) g2 (h1 h2–1)–1 Note that in general, (ab)–1 = b–1a–1

Yes. And taken over all elements h of the group, the resulting set is an equivalence class called a conjugacy class. For a given h, the mapping g –> h g h–1 is called an inner automorphism. Subgroups of the group that are invariant to all inner automorphisms of the group are called normal subgroups of the group. Normal subgroups are very important in group theory. Note that all subgroups are normal in abelian groups.

In group theory, the elements h g h–1 for each and every element h of the group form an equivalence class for element g of the group. If g belongs to the centre of the group, then its equivalence class is g alone. Otherwise, the equivalence class contains multiple elements. Thus, although the equivalence classes of a group partitions the group, the equivalence classes are not necessarily equal in size.

Can you provide a proof? I can't at this time provide a counterexample. EDIT: I think you may be correct. If one chooses a single non-identity element as a generator, then if it is a finite generator, the subset generated must contain the identity element and all the inverses of the elements of the generated subset, and hence be a group. But if the generator is not finite, then the subset generated is not necessarily a group.

No. The subset also must contain the identity as well as the inverses for all elements of the subset. The associative property is however inherited from the group G.

Humans use technology to modify the genetics of various organisms. Is this part of evolution or separate from it?

I believe so, yes. All elements equivalent to a given element are equivalent to each other. Thus, the set X is partitioned into subsets of elements that are equivalent to other elements of the same subset but not equivalent to any elements of the other subsets.

Why do we need to witness the entire "reel" simultaneously? You also said: Even if that were true, so what? Why is it so important that things be directly visible? What is wrong with calculated results?

I don't know if it can be considered to be "proven", but quantum mechanical outcomes are intrinsically random.

I don't know what resources you have, but one way to distinguish between ethanol and 2-propanol is to oxidise the sample with excess acidified dichromate or excess acidified permanganate under mild conditions. Ethanol will oxidise to acetic acid, whereas 2-propanol will oxidise to acetone. These are more readily distinguished than the original alcohols, although the practicalities of doing so might not be straightforward. However, the difference in odour may suffice.

Not everything is relative. For example, the "distance" between any two points along a given path in spacetime is invariant, the same for all observers, the same in all coordinate systems. Length contraction and time dilation occur because the "distance" measured in different frames of reference isn't between the same two points along the same path in spacetime.

https://phys.org/news/2024-12-dark-energy-doesnt-lumpy-universe.html "One of the biggest mysteries in science—dark energy—doesn't actually exist, according to researchers looking to solve the riddle of how the universe is expanding. Their analysis has been published in the journal Monthly Notices of the Royal Astronomical Society Letters." Basically, they claim that there is no need for dark energy because the apparent accelerated expansion of the universe is really due to how we calibrate time and distance.

I didn't put the two statements together to logically connect them. I put the two statements together because individually they are both begging the question. The first statement is an observation grounded in scientific evidence of what exactly? Point #5: "5) A pertinent question arises: is it merely coincidental that the most suitable environment for life to emerge and thrive happens to be the same environment we inhabit? This coincidence, though seemingly extraordinary, is a reflection of the conditions under which observers, such as ourselves, can arise to question it." seems to me to be a statement of the anthropic principle. This statement from point #6: "Our very existence serves as empirical evidence that such an event, however improbable, has occurred." seems to me to be saying that our existence is evidence that the probability of forming life was not too low for it to occur without an intelligent designer, whereas this is the conclusion at which you are trying to arrive and thus begging the question. Your opponent could just as easily say that our very existence serves as empirical evidence that life occurred by an intelligent designer. Our existence may be empirical evidence, but of what exactly? With regards to the anthropic principle, I posted (on another forum) the view that the multiverse of all possible universes combined with the anthropic principle guarantees our existence regardless of how improbable life is (assuming that it is not actually impossible), making the point that our existence says nothing about its likelihood. At the time, I was suggesting that life was too improbable for it to exist elsewhere in the observable universe. Such a low probability wouldn't affect the likelihood of our existence but would affect the likelihood of the existence of alien life. The reason I mention this here is because the anthropic principle alone doesn't guarantee our existence in the absence of an intelligent designer, but combined with a multiverse, an intelligent designer is no longer necessary regardless of how improbable life is. One additional remark I'd like to make is that I don't think the complexity of life can ever make it impossible, only improbable.

I'm inclined to agree with you on this point. There does seem to be the view that the laws of physics are arbitrary as if given to us from above, a view which I personally reject. My own view is that by constructing a description of reality, one now has a mathematical object whose properties can be established by mathematical means. And because of the correspondence between reality and its description, the mathematical properties of the description will correspond to laws of physics.

Why do you think that doing maths is some magical thing? Did it ever occur to you that the underlying basis of the laws of physics may be mathematical?

With these two statements, you are begging the question.