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His problem is with mathematics that can't be written down. He sees that a lot of modern mathematics, especially everything to do with real numbers, deals with objects that you can't actually write down or manipulate directly. Instead of establishing actual rules on how to perform operations on them, you take it as an unproven axiom that these operations can be performed. In reality in order to perform these operations in general it takes an omniscient being, basically God. So in this way mathematicians resort to using indirect, roundabout language like "Let x is the result of performing this impossible operation" without ever actually exhibiting x. This, in his mind, takes away from the beauty and precision of mathematics. Mathematics is no longer manipulating objects and formulas in smart ways but rather it becomes talking about what would happen if we could manipulate it in ways that no finite being could. A shallow view into the problem might suggest that the main trouble is with the axiom of choice, which asserts the existence of objects without you needing to exhibit them, but Wildberger realizes the trouble runs much deeper than that. According to him, the main cause of the difficulties is in the imprecise use of terms in mathematics. If you adopt this approach, if you require your terms to be defined clearly and precisely, you will be lead to finitism.
A common misconception is that finitism is about rejecting infinities of all types. This is not true. Wilberger is completely fine with infinities as long as they are defined precisely and the rules for how to manipulate them are laid down. An example of this is the infinite points in projective geometry, a topic he has done numerous videos about. These infinite points can be defined clearly and precisely so that no ambiguity arises.
