A Tripolation

Uh, what? How does GMO implementation lead to this at all? You realize that if us farmers can't get ahold of GMO seeds, we'd just plant regular seeds, right?

To begin with, I'm in my first real mathematics course this semester. Most of my work has been concentrated in physics and I have no background in analysis and only some basic set theory. The question is: Let [latex] R^\infty[/latex] be the set of all infinite-tuples [latex](x_1, x_2,...)[/latex] of all real numbers that end in an infinite string of 0's. Define an inner product via [latex] \left\langle x,y \right\rangle = \sum_{i=1}{^\infty } x_i y_i[/latex] Let ||x-y|| be the induced metric on [latex] R^\infty[/latex]. Let [latex] e_i[/latex] be the vector with a 1 in the ith entry and a 0 in the other entries. These form a basis for [latex] R^\infty[/latex]. Let X be the set {[latex]e_i[/latex]}[latex]_{1\leq i \leq \infty}[/latex]. Show that X is closed, bounded, and non-compact. I understand the formulation of the summation for the dot product and the formation of the basis using those vectors (it forms a linearly independent set that spans the space, IIRC my algebra correctly), but I don't know where to start showing that this set is closed. Or bounded (why would an infinite set be bounded?). I have some notes on compactness, but the concept is quite foreign to me. Any further explanations on the topic would be quite beneficial to me. Thanks!

Morals aren't real. Case closed. You're a really sucky atheist.

Yes. If something named an "apple" didn't exist, their tree would not exist. You are completely correct! Welcome to logic 101. Another example: if a mathematical operation called the "additive inverse" did not exist, it's function on math, a subtraction, would not exist! True! Religion is a categorical name given to a set of epistemological frames and beliefs! There exists no ontic entity known as "religion." It is a social construct. Nothing more. I actually explained where the science of morality comes from- repeated prisoners' dilemma problems and such. So you agree that morality is nothing more than applied neurochemistry to social situations. Good to know! You don't think it exists, either. You simply don't see that yet. Now, can you explain how you got from"morality can be shown to be an outcome of intelligent social interaction between animals" to "To suggest that morals are something that can exist without social creatures manifesting it shows an absolutely abysmal grasp on philosophy. " I see nothing mutually exclusive about these statements. This actually has nothing to do with my religion since I'm quite the deist. This has to do with your knowledge of the philosophy of morality being so close to zero that the difference is negligible. Humans decided (most of them did) that the pain and suffering of a sentient, enslaved human wasn't worth what they got from it. Slavery has been rampant in human history and it was mostly an acceptable practice. Much like claiming lands from your defeated foes was an acceptable practice until the early 1800s. Social beliefs change. I expect a similar change in the future when humans become almost completely vegetarian in that there's no sense killing an animal if you derive no additional nutrition from it than you would a plant-based diet. Right now, being a carnivore is a morally acceptable position. This might change in the future. Morals change based on the society you live in. There is no absolute morality.

Gravity is the curvature of spacetime caused by mass-energy. Removing mass only removes a very tiny and very insignificant source of mass-energy. So, no. That's a horrible analogy. Remove ALL of the mass-energy in the universe then I will say gravity doesn't exist. Yes, altruism and the functions selected by group theory and evolution exist in other animals with social hierarchies. This does nothing but prove my point that morals are a fabrication. They ONLY exist in social creatures. Does a star have morals? Does a black hole feel guilty as it accretes matter away from its binary companion, resulting in its demise? No. To suggest that morals are something that can exist without social creatures manifesting it shows an absolutely abysmal grasp on philosophy. Morals aren't real. It's a label for certain actions that result in an increased rate of survival for a group. Also, I'm going to say this: if you're suggesting moral absolutism is the case, then that absolute morality needs to come from somewhere. That source must be a perfect source of good morality, because all "evil" is is the deviation from good morality. This source is almost always God. Taking a stance on morality like yours makes it almost impossible for you to refute the existence of an omnibenevolent entity. Again, you should really look at the science of the matter.

They are detrimental to the human organism as a whole, yes. That's why group theory and evolution don't really select for that. If you think this in any way proves morals are separate, ontic entities, you're insane. Morals are social constructs. They would not exist without humans. They are not truths. They are not logical axioms. They are not derived.

If that were true, there'd be a universal code of ethics every society would follow. This is clearly not the case. Morals differ by culture.

Said the atheist to the theist. You really might want to look into the philosophy of morality. It wasn't wrong at the time. And, seriously, this conversation can't produce any results. I start with the idea that morals aren't real. You think they are. There can be no constructive discourse.

Slavery is wrong *now.* It wasn't wrong, then, according to their beliefs. Morals are constructs of the society we live in, the people that exist, and behaviors instill by evolution. There is no absolute morality. To say such a thing exists is wholly unscientific and very poor philosophy. That the world is round is an objective, empirical, scientific fact. To say that slavery is wrong is a temporary, moral "truth." Conflating the two earns you no points in the matter. It doesn't change the FACT that slavery was an accepted social mores at the time.

This adds absolutely nothing to the discussion. I have no idea how this is a rebuttal to my last post.

It was true at the time. A 13-year old boy was considered a man grown, but that's not true now, is it?

I don't think anything in the OT is really what God thinks. Not many people do. The NT is a lot closer, but it's still flawed. It's still just a recounting of stories passed down orally of a tumultuous time. Now, had God written the OT personally, then there might be a problem. But AFAIK, it was written by warring pastorialists.

*gasp* Slavery is bad?! Thank you so much for letting me know! The level of dissonance coming from you right now is astounding. no one is claiming that slavery is good. But, for that time period, it was completely normal to own slaves. In what universe does "authors of an ancient text held a position that isn't morally tenable" equivalent to "religion does not always become a cognitive burden, but, rather an asset"?

Quantum mechanics is a theory with acausal mechanisms and it is quite sound. I don't anticipate it being turned over that its tenets will need to be rewritten, but it is possible. Highly unlikely, though. And I thought this was the KCA thread. My bad. I was using it as a direct counterexample of "everything that begins to exist has a cause," which is most certainly not true at the quantum level. The FTA is destroyed when you realize that it is possible for other universes to form with radically different fundamental constants. This is not the only valid universe and it is egocentric to think so. There are many valid configurations and if the MWI of QM is correct, then there are infintely many valid universes and FT would be quite a silly statement.

That's not really an explanation, nor is it a cause. Here's how Swansont explained it to me back when I first took quantum mechanics. //It's causeless in the sense that there is no trigger, i.e. it's spontaneous — it doesn't require an outside event to cause it. This is supported by the random decay time, which indicates that it's a probabilistic event. Non-spontaneous events require an outside influence of some sort — some particle that undergoes an interaction. Decays lack this. Seeking the lowest energy is a true; classically we see this as a force of some sort. But those events happen immediately. A ball on an incline rolls downhill. It doesn't wait an arbitrary length of time before doing so. For alpha decay, it's the alpha hitting the potential barrier, and having a chance to tunnel through, but tunneling is probabilistic. Nothing causes the tunneling to occur . "Random" is not a particularly difficult model. Random fluctuations in the nucleus are not caused, so at best you've moved the explanation one step back.//