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Evolution is a testable, falsifiable, natural explanation to physical phenomenon. That makes it science. Creationism is an untestable, unfalsifiable, supernatural explanation to physical phenomenon. That makes it religion.

How do you test the past?

The same way you test anything in science. You make predictions based on the theory and test them. Think of it as a series of "if/then" statements. "If this aspect of evolutionary theory is correct, then we should see this under these conditions."

An example of this that has been used to test one aspect of past evolution is:
If this group of animals is more closely related to each other than to other animals, then we should see that their DNA sequences are more similar to each other than to other animals (based on what we know about how DNA is passed down with modification).

And indeed, that is what we see.

Do cats and dogs have sex and make dog cats?

No. And evolutionary theory doesn't predict that. In fact, evolutionary theory explains why they don't.

Why not? If we all evolve from the same place, then why not? Cats and dogs cannot come together and produce kitten puppy hybrids? What did Charles Darwin say in his 1859 book Origin of the Species?

Because of evolution. Weird that you are trying to argue with me when you clearly don't understand it. Evolutionary theory predicts that as populations of a species interact less (think about two groups getting separated or traveling in different directions or living in different places) genetic differences we begin to accumulate between them. The longer they are separated the more different they will become. Eventually they will become so different they will no longer be able to breed with each other. Then you will have two different species, which arose out of one. That is how species originate, or what Charles Darwin called "The Origin of Species".

That is a fallacy of affirming the antecedent (I think). It’s not enough to say "if X is true then we predict Y; we see Y therefore X is true." This approach fails to acknowledge any other possible explanations for Y.

All you can say logically is that if Y is true then X MAY be true.

Another very obvious explanation for the scenario you offer is that DNA sequences are similar because the animals have the same designer.

With the logic you offer, one can prove anything: flat Earth, life is a simulation, or red is blue.

It's not a fallacy, and no one is saying "we see Y, therefore X is true." We are saying, if X is true we should expect to see Y. And we do that over and over and over again, for different predictions. And we evaluate predictions for alternative hypotheses as well. For evolution this has been going on for 150 years and everything we have learned about biology in that time has affirmed evolutionary theory. That gives us a strong indication that it is correct. Yes, it could also be a designer, but in the case of a designer you could say it was designed that way no matter what you see, which is why I say it is not falsifiable and therefore not science.

Ok, I stand corrected on the fallacy thing, provided you are not actually standing on these predictions as proofs.

But to say that everything we have learned about biology for the last 150 years affirms evolutionary theory can only be true if you are wearing a particular color of glasses. There are many biological processes and constructs which evolutionary theory cannot explain except with the most optimistic hypotheses. I have never heard of a good refutation of irreducible complexity, for example. There are myriad elements in biological systems which would have no use and no contribution to the survival of the fittest by themselves, and many systems which can only work if all the elements are first in place.

No, it is not.