 People love to cling on to their theories about how the world works. But if you're interested in the world of ideas, you've probably noticed that orthodox standard theories are often times fundamentally wrong. So how do you go about presenting a superior alternative to a dominant theory? Well, I think there are three things that have to be accomplished if you want to replace one theory with another. I'll tell you what the three things are and then I'll give you an example. The first thing that has to be done is you have to demonstrate where the shortcomings are of a particular theory. That means you have to demonstrate what phenomena is not explained by a particular theory or you have to demonstrate some kind of conceptual confusion or internal incoherence in a particular theory. Two, you have to demonstrate where your theory or the unorthodox theory solves those problems. So you present the problem and you solve the problem by presenting an alternative theory. That is not enough. This is the one that people always overlook. The third thing that absolutely must be accomplished is you have to explain in your theory why the theory you think is wrong had predictive power. Why did the theory that you think is wrong, why did it work so well? So let me give you my favorite example, the Ptolemaic model of the solar system which says the earth is in the center of everything and then all the planets and all the stars revolve around the earth in a perfect circle. It is a beautiful theory. It has incredible predictive power. The Ptolemaic model of the solar system can satisfactorily explain the motion of the stars. So why would somebody move from the Ptolemaic model or the geocentric model into something like a heliocentric model thinking that maybe the sun is the center of the solar system? Well, what has to be done first is you have to demonstrate where the theoretical shortcomings are of the dominant theory. So in this case, if you observe the changing phases of Venus, that is not satisfactorily explained in the Ptolemaic model of the universe. So we have a problem with the theory. Ah, that's the problem. Then two, you have to demonstrate why your proposed theoretical alternative is superior. You have to satisfactorily explain the changing phases of Venus. What's one way that you can do that? Well, you can say, hey, if the sun is in the center of the solar system and everything's revolving around the sun, well then that also has predictive power. It can explain the motion of the stars and we don't have this problem with the changing phases of Venus. And then number three, you have to explain why the Ptolemaic model had such predictive power. And in fact, you can do so. You can say, okay, look, I realized that theory worked. It worked really, really well, but there was just these edge problems and because of these edge problems that couldn't be accounted for, we have to rethink the entire paradigm for how the solar system operates. Earth isn't in the center of everything, even though that gives you predictive power. The sun is in the center of the solar system. So why does this matter? Well, as I'm hinting at consistently and it will be more clear over the next year or so, this is what must be done, at least in my work, in mathematics. I'm claiming that there are foundational mistakes in the way that mathematicians have been conceiving about mathematics for quite some time, but especially in the last century. So I have to explain, in this case, the conceptual problems with the traditional way, the standard orthodox way of thinking of mathematics. Two, I have to demonstrate where there's a superior theoretical alternative. So where finiteism solves the problems that are presented with the modern paradigm of thinking about mathematics. And then three, I have to explain why the modern way of doing mathematics works. So concretely speaking, there are conceptual problems with calculus. People think that in order for calculus to work, it has to include actual infinities. So what I have to do is say, no, it can't include actual infinities. There's conceptual confusion. Two, I have a theoretical alternative that preserves the explanatory power of calculus and then resolves the conceptual difficulties. And three, I still can explain why the inaccurate way of thinking about calculus works. So that's something I've done just a tiny little bit in some of my writing, but I'll continue to do in these videos and other pieces of writing. So if you are a thinker in some particular area and you think the dominant paradigm is wrong, you have to accomplish these three goals. Where is it wrong? Why does your resolution solve those problems? And then why has the dominant paradigm been so successful?