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Originally Posted by Mightyboosh
Thanks for the answer. I understand the line you're taking there but lack the skills to understand the maths, so I'll have to take your word for it for now. I won't be able to use it because I couldn't defend it, although I could share it I guess, and see what others make of it.
Sorry I do tend to assume others know what I do when I post and that probably was unnecessarily opaque to a non-math geek. Think of it this way. Suppose you deposit $100 in some kind of investment account that pays 7% return. The math says that every 10 years the value of your account will double. In 10 years you have $200. In 20 years you have $400, in 30 years $800 and so on. This is the fundamental property of what is called an exponential growth function. Any time that the rate of change in a given quantity depends on the current value of that quantity you get this kind of growth. For an investment account, the amount of change in the value during any given year is equal to 7% of the current value, so this type of growth applies.
Now realize that the math doesn’t care whether you run time forward or backwards - the same fundamental property exists. If you look at your account value 10 years from now it will be double what it is now. If you look at the value 10 years ago, it was half the current value. 20 years ago it was 1/4 of its current value, 30 years ago it was 1/8 of its current value, and so on.
The analogy now breaks down because money is not infinitely divisible (we can’t have less than $.01 in our account), but real observables can take on smaller and smaller values. No matter how far we go back, the value of our “account” can never reach zero. It can become extremely close to zero, but can never actually get there. Take any number and divide it by 2 repeatedly. It will get closer and closer to zero as you keep dividing, but it can’t ever actually get there. That is the crux of the argument- the constant increase in entropy implies a low entropy state in the past, but it does not necessarily imply a zero entropy state at any past time.
Note that this is perfectly consistent with a constantly increasing entropy. The second law of thermodynamics says entropy is increasing, but does not say by how much it is increasing at any given time. If you go from one very small almost zero number to a second very small, almost zero number that is greater than the first, that is still an increase, even if it is a very small one.