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Oh, yes, it is *any* object in the category. So, the point of having the arbitrary object and the 'extra' processes is to ensure that the product looks the same to every object in the category? To answer my own question from the OP, if X were R (since we know R is an object in our category), then f1=f2=I and f=p1-1=p2-1. That section is on neither the Amazon Look Inside nor the Google Books preview.

When we're talking about the teachings of Jesus, we should probably listen to Jesus rather than Paul. Jesus was very explicit. He said when he'd fulfill the Law and he that time hasn't come yet. He says he'll fulfill it when heaven and Earth pass away. The OT Law is valid to the letter until the end of the world. You're off the hook after the second coming is over. Think about it, if Jesus already fulfilled everything, what's he coming back to do? Or we can just read what he said, since salvation by works was one of his main points. "Ye are the salt of the earth: but if the salt have lost his savour, wherewith shall it be salted? it is thenceforth good for nothing, but to be cast out, and to be trodden under foot of men. Ye are the light of the world. A city that is set on an hill cannot be hid. Neither do men light a candle, and put it under a bushel, but on a candlestick; and it giveth light unto all that are in the house. Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven. Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach [them], the same shall be called great in the kingdom of heaven. For I say unto you, That except your righteousness shall exceed [the righteousness] of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven."-Matthew 5:13-20 "Enter ye in at the strait gate: for wide [is] the gate, and broad [is] the way, that leadeth to destruction, and many there be which go in thereat: Because strait [is] the gate, and narrow [is] the way, which leadeth unto life, and few there be that find it. Beware of false prophets, which come to you in sheep's clothing, but inwardly they are ravening wolves. Ye shall know them by their fruits. Do men gather grapes of thorns, or figs of thistles? Even so every good tree bringeth forth good fruit; but a corrupt tree bringeth forth evil fruit. A good tree cannot bring forth evil fruit, neither [can] a corrupt tree bring forth good fruit. Every tree that bringeth not forth good fruit is hewn down, and cast into the fire. Wherefore by their fruits ye shall know them. Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven. Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works? And then will I profess unto them, I never knew you: depart from me, ye that work iniquity. Therefore whosoever heareth these sayings of mine, and doeth them, I will liken him unto a wise man, which built his house upon a rock: And the rain descended, and the floods came, and the winds blew, and beat upon that house; and it fell not: for it was founded upon a rock. And every one that heareth these sayings of mine, and doeth them not, shall be likened unto a foolish man, which built his house upon the sand: And the rain descended, and the floods came, and the winds blew, and beat upon that house; and it fell: and great was the fall of it."-Matthew 7:`3-27 "Then said Jesus unto his disciples, If any [man] will come after me, let him deny himself, and take up his cross, and follow me. For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it. For what is a man profited, if he shall gain the whole world, and lose his own soul? or what shall a man give in exchange for his soul? For the Son of man shall come in the glory of his Father with his angels; and then he shall reward every man according to his works. Verily I say unto you, There be some standing here, which shall not taste of death, till they see the Son of man coming in his kingdom."-Matthew 16:24-28 "And, behold, one came and said unto him, Good Master, what good thing shall I do, that I may have eternal life? And he said unto him, Why callest thou me good? [there is] none good but one, [that is], God: but if thou wilt enter into life, keep the commandments. He saith unto him, Which? Jesus said, Thou shalt do no murder, Thou shalt not commit adultery, Thou shalt not steal, Thou shalt not bear false witness, Honour thy father and [thy] mother: and, Thou shalt love thy neighbour as thyself. The young man saith unto him, All these things have I kept from my youth up: what lack I yet? Jesus said unto him, If thou wilt be perfect, go [and] sell that thou hast, and give to the poor, and thou shalt have treasure in heaven: and come [and] follow me. But when the young man heard that saying, he went away sorrowful: for he had great possessions. Then said Jesus unto his disciples, Verily I say unto you, That a rich man shall hardly enter into the kingdom of heaven. And again I say unto you, It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God."-Matthew 19:16-24 "When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit upon the throne of his glory: And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divideth [his] sheep from the goats: And he shall set the sheep on his right hand, but the goats on the left. Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world: For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me. Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed [thee]? or thirsty, and gave [thee] drink? When saw we thee a stranger, and took [thee] in? or naked, and clothed [thee]? Or when saw we thee sick, or in prison, and came unto thee? And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done [it] unto one of the least of these my brethren, ye have done [it] unto me. Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels: For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink: I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not. Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee? Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did [it] not to one of the least of these, ye did [it] not to me. And these shall go away into everlasting punishment: but the righteous into life eternal."-Matthew 25:31-46

I'm not sure I understand exactly the product in category theoretic terms. The book I'm using gives a very concrete example that is easy to understand. However, I'm not sure I get it generally. How does it work for the product of the reals? What's X in that case? What are f1 and f2?

He did? I wasn't aware that the end of days had already happened. When, exactly, did the end of the world happen?

Maybe something like this:

! Moderator Note Off-topic speculations post hidden. Keep speculations out of mainstream sections.

You what, you're right. My bad. k does increase. Good catch.

Let's look at the energy. You start with a compressed spring. This spring uncompresses. In uncompressing, it releases energy to the environment in the form of heat. If you don't believe me, grab a paperclip and bend it back and forth a few times. This powers the generator. The generator shaft releases energy to the environment in the form of friction heat and noise. So, the energy going to the electrical generation is lower than what we started with. E0>Eshaft The electrical generation gives off quite a bit of energy in the form of heat in both ends. The wire connecting the generator to the motor also gives off heat. So, we're down even more. Eshaft>Emotor input The electronics of the motor give off heat. The shaft of the motor gives off heat and sound. So, Emotor input > Espring input Compressing the spring releases heat. Again, grab a paperclip. Espring input > Ef So, for one cycle, we get E0 > Ef. The system loses energy in even one cycle. Now, the story gets even worse for your device, because the compression and uncompression of the spring alters the material properties of the spring. The amount of energy in a compressed spring is a function of not only the distance compressed, but also the material properties of the spring. As the spring goes through compression cycles, it becomes less springy and so holds and requires less more energy for the same compression distance. So, not only does the system lose energy every cycle, every cycle makes the spring able to hold less and less more and more energy to power the generator. And that's without the system taking on a load which would remove more energy. So, if you want your machine to *do something*, then you need to account for that loss too. Where are you imagining the extra energy comes from? Actually, it is. The energy to compress a spring, and released by the spring, is given by E=(1/2)kx2. The act of compressing and uncompressing changes k. k1>k2, so Efrom release>Eto recompress. This, however, means the spring can hold less energy as well. As the spring cycles tend to infinity, k tends to zero. And, if it cycles fast enough, it will melt first. As pointed out by studiot below, k goes up instead of down. EDITS ARE IN ITALICS

! Moderator Note Relinking post deleted. DO NOT link it again. Do not reply to this modnote.

I agree with this. Why should we start with values rather than something more general like sets (or even more general than that, and use categories)? You can start with objects A, B, and C, as well as processes x:A->B and y:A->C [A]-> | \/ [C] And then you can define general division in terms of sections and retractions. If you're wanting to find foundations (which, as a philosopher in an HPS program, I think are unlikely to be found), then shouldn't you go as generally as possible? If you want to use logic, addition is actually defined as the disjunction of sets, not their conjunction. Zero is the null set. One is the set containing only the null set. Two is the null set disjoined the set containing only the empty set (or the set whose only two members are the empty set as well as the set containing nothing but the empty set). Three is the set whose three members are the null set, the set containing only the null set, and the set containting only the null set and the set containing only the null set. And so forth. 0={}, 1={{}}, 2={{}, {{}}}, 3={{}, {{}}, {{}, {{}}}}, etc. It sounds like what you want here is the Peano axiomatization.

! Moderator Note I would say so.

In the Alcubierre drive, the area where the ship is is flat. There's no time dilation; the clock ticks the same as it would in deep space.

I'm not saying it's aliens, but it's definitely aliens.

So, basically, just do that backwards?

Is there a way to get a quantitative prediction out of a literal reading of the Old Testament. The Young Earth Creationists hold that the OT is literally true. The human race started with two individuals roughly 6000 years ago. 4000 years ago, the resulting population was culled to a family of 8. That family of 8 gave rise to everyone alive now. Is there a way to see quantitatively how much genetic variation this would allow? If so, how would one go about doing it? I'd like to see how it compares to the observed variation.

There's a really cool reason why we can't have perpetual motion machines with a load. They put out more energy than they take in. You might be thinking, "So what, energy conservation is wrong". But here's the kicker; energy conservation, rather than being an assumption simply falls out of the general way in which the laws of physics are constructed. Without boring you with the equations themselves, the difference between potential and kinetic energy has a strict relationship to both position and momentum. From the equation that specifies this relationship, Emmy Noether was able to prove a few theorems. One of them related certain kinds of symmetry with conservation laws. Time translation symmetry just so happens to imply conservation of energy. So, if your motor worked, we would literally have to rewrite all of physics.

By the way, it's not even clear whether quantum gravity must have any issues with black holes. Loop QG, for instance, removes the singularity by showing that the BH collapse is a bounce and that the inner volume never even reaches the planck scale; it happens exceedingly quickly, but looks like it is static due to gravitational time dilation.

Even if your recollection is correct, Kaku on TV is anything but a reliable source. He is notorious for spouting nonsense that he knows is false simply to make some $$$ from a TV spot.

Again, thanks. This was much faster than an amendment.

Do tell.

Huzzah! I'll take five.

Will it be a picture book?

Get out a pen and some paper. Draw the function f(x)=1/x. Don't just do the positive x's. Do the negative ones as well. Tell us what happens at x=0.

Too bad the US doesn't officially keep track. We have to cobble together numbers from news stories.

It's a very simplified introduction to my field. If you liked it, consider taking History and Philosophy of Science as your major in college.