In a new video, Professor N J Wildberger calls for the end of ‘infinity’
Norman J Wildberger is an Associate Professor in mathematics at UNSW (University of New South Wales) in Sydney Australia, and he has a PhD in mathematics from Yale. He is well-known for his excellent educational videos on YouTube and for being an Ultrafinitist. He voices some of his objections to the concept of ‘infinity’ in his recent video (below) “MathFoundations178: The law of logical honesty and the end of infinity”.
At 27 seconds into the video he announces “It’s time for us to face the music and expel infinity from mathematics”. He goes on to consider the physical universe and argues there are no “infinite things” in our world. He claims we cannot imagine an ‘infinite’ set, we only imagine we can imagine one. If we accept the existence of finite sets, does this mean other sets must exist that are ‘not finite’? Surely not, because if we can see a visible hotel it doesn’t mean an invisible one must exist, and if we have a moveable kitten it does not mean an immovable kitten must exist.
There is an argument that sets which are not finite must be infinite. This presupposes that both finite and “infinite” sets already exist, whereupon we could say the ones that are not finite are “infinite”. Professor Wildberger says “we don’t have such a pool of sets to begin with”.
Regarding the removal of infinity from mathematics, he warns “This is an argument that has enormous implications” and “a lot of mathematics really has to be re-thought”
You can watch the video below.
I (Karma Peny) posted a YouTube comment to highlight the issue that we have no clear statement to describe what mathematics is. I said that mathematics was originally devised to solve real-world problems and that it was underpinned by real-world physics. It’s apparent generic nature can create the illusion that mathematics has its own ‘existence’ and that it is not simply a tool based on real-world physics.
I compared the axiom ‘an infinite set exists’ with an axiom ‘the god of thunder exists’. We can claim both are consistent and cannot be disproved, but both these axioms are equally worthless and irrelevant in the real-world, just as are any deductions derived using these axioms. I concluded “It is often argued that the use of ‘infinity’ in mathematics has proven to be very successful, but the successes could be despite the use of ‘infinity’ rather than because of it. I suspect we will have more clarity and even more successes if we abandon the use of non real-world axioms”.
My comment attracted a brilliant response from YouTube user ‘Amonojack A’. Here is part of this response… “So-called ‘pure math’ was born out of the idea that it might be worth developing mathematical objects and relations that correspond to no physical situation yet discovered, but could. Seems noble enough. The problem came when people failed to keep track of context. They floundered into musing about things that not only had no known physical analog, but that couldn’t ever even conceivably have a physical analog”.
Professor Wildberger was so impressed with these comments that he published them both in their entirety here, on his own website.