These Mathematicians Don’t Believe Large Numbers Exist. I’m Serious.

Infinity comes up in physics everywhere

and yet it's not physically real. This

is why some physicists have argued that

really our problems in the foundations

of physics might come from the way that

we calculate with the infinitely large

and the infinitely small. I just

recently learned that they now have

backup the ultra finitists.

That's mathematicians who deny the

existence of large numbers are not

making this up guys. Might this be the

paradigm shift we've been waiting for?

Let's have a look. The universe is

infinite in our mathematics. The

curvature at the big bang is infinite.

So is the gravitational tidal force in

the middle of a black hole. We have

infinities in quantum physics too all

the time. When we calculate something,

we have to do mathematical gymnastics to

get rid of those infinities. The

cosmological constant, some physicists

think, is a relic of our problems with

infinity. And the infinitely large has a

flip side. That's the infinitely small.

Space and time are made of infinitely

many points of size zero. At least in

our mathematics. This isn't just the

case for space and time. It's also the

case for quantum physics. In quantum

physics, we work with wave functions in

a hilbert space. Don't worry if you

don't know what this is. It doesn't

matter all that much. You just need to

know that Hilbert space 2 is made of

infinitely many points of size zero.

It's a continuum. In the past decade,

we've seen several physicists rebelling

against this abundance of infinities.

The cosmologist George Ellis for example

thinks that careless use of infinity is

the reason why physicists ended up

believing the multiverse is real. We

have also previously talked about

Nicolola Jiza who thinks that we

shouldn't work with real numbers because

real numbers have infinitely many digits

after the point which requires infinite

precision and that's something which

doesn't exist. There's also Tim Palmer

who says that all our problems with

quantum mechanics come from using

Hilbert space with its continuum. He

says we need to discretize it and take

only some wave functions in this space.

It's like a return to the early days of

quantum physics where the word quantum

originated in discretetness. Finite

steps, no infinities, no zeros. But most

physicists ignore these ideas. They

believe that using infinitely large or

infinitely small numbers is just a

mathematical tool, an approximation.

It's innocent. We could make a lot of

effort to get rid of it, but in the end,

the result will be the same. So why

bother? What really is the difference

between 10 to the 100 and infinity?

Well, that's the question, right? Will

the result be the same? or do we

actually get something wrong when we use

infinity and we don't notice? I don't

have a good answer to this, but I think

it's a fair question to ask. And this

brings me to the ultra finitists. This

is a philosophical movement of people

who not only want to get rid of all

infinities in physics or in the maths

that we use in physics, but they also

want to get rid of all really large

numbers. Concretely, Joe Humpkins, a

maths professor at the University of

Notra Dam in Indiana says, "According to

ultra finitism, the various extremely

large numbers mathematicians

conventionally take themselves to

describe, such as 10 to the 100, do not

actually exist. The new thing is now

that mathematicians are trying to figure

out how to define mathematics using only

small numbers like consistent ultra

finitist logic or bounded arithmetic

especially when it comes to complexity

bounds that is the question of what you

can or cannot compute and of how

complexity grows. They had an entire

conference on this just in April. So the

ultra finiteist philosophy has this

mathematical backbone and this in return

might impact physics. It could explain

for example why entropy seems to have a

maximum bound that black holes fulfill.

It could explain why we have infinities

creeping up on us in quantum theory. Or

maybe it could even help us quantize

gravity by doing away with the

infinities that we keep running into.

Maybe this is the tool we need to make

progress in the foundations of physics.

Let me be honest though. I'm not sure I

understand what it means to say that a

number like 10 to the 100 doesn't exist

like I mean I can write it down and

compute with it. So how does it not

exist? I can understand that physically

certain quantities like say the number

of protons in the universe might be

finite but denying the existence of

numbers that's another level of

denialism entirely. So I don't see the

point but I'm trying to be open-minded

about these things. Maybe there's

something to it that I can't see. And in

any case I see it as my job to keep you

informed regardless of whether I

approve. So here you go. Number deniers

next week. Rectangles don't exist. How

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tomorrow.