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Stream: deprecated: mathematics

Topic: mathematical phantoms

John Baez (Dec 06 2020 at 21:58):

A 'mathematical phantom' is a mathematical entity that does not literally exist, but acts as if it did. Here I list some example and call for more:

• Mathematical phantoms.

Matteo Capucci (he/him) (Dec 07 2020 at 11:04):

One of the coolest posts of the n-Cafè so far! I really love phantoms. Some of my favorites mentioned in the post or in comments so far

• uniform probability distribution on the real line/natural numbers
• infinitesimals
• field with one element
  As you also noted, John, phantoms are often the telltale sign that something is amiss in our understanding, i.e. we are working at a sub-optimal level of generality.
  A beautiful example of this are negative and complex numbers, which were phantom numbers (useful to think about equations, but not 'real') for quite a bit.

Fabrizio Genovese (Dec 07 2020 at 12:37):

Can we really consider infinitesimals to be still phantoms? A huge amount of work has been done in formalizing them and I'd consider them 1st class citizens as of now...

Fabrizio Genovese (Dec 07 2020 at 12:38):

Another phantom could be a purely equational presentation of field axioms. This has lead to interesting definitions, such as meadows.

Nathanael Arkor (Dec 07 2020 at 12:45):

"Small (nonposetal) cocomplete category" has a similar flavour to some of the examples mentioned here (but may actually be reified in a constructive setting). The study of locally presentable categories is in part inspired by this search. Are there any other categorical examples?

Matteo Capucci (he/him) (Dec 07 2020 at 12:57):

Fabrizio Genovese said:

Can we really consider infinitesimals to be still phantoms? A huge amount of work has been done in formalizing them and I'd consider them 1st class citizens as of now...

Indeed, as I pointed out in my comment on the n-Cafè they're only phantoms in 'mainstream mathematics' though, since you usually work in a setting where infinitesimals are not real numbers. In particular, I'm pretty sure most analysts do not know you can actually formalize them, while a tiny percentage may have heard of non-standard analysis à là Robinson and an infinitesimal (pun intended) fraction knows at least the basics of SDG

Fawzi Hreiki (Dec 07 2020 at 12:58):

They used to be phantoms but now they aren’t I guess you could say

Fawzi Hreiki (Dec 07 2020 at 12:59):

But in a sense, all phantoms are ‘proper mathematical objects waiting to be born’

Fawzi Hreiki (Dec 07 2020 at 12:59):

It just takes a shift of perspective

Matteo Capucci (he/him) (Dec 07 2020 at 13:00):

In fact, I'd say pure phantoms do not last long since once you realize you may want them you're halfway through fomalizing them. A different kettle of fish is to formalize them right. E.g., $\Bbb F_1$ has been formalized for at least 25 years but we are still in the pre-paradigmatic phase of the associated Khunian revolution.
Infinitesimals are probably stuck there too.

Amar Hadzihasanovic (Dec 07 2020 at 13:01):

So mathematical phantoms go the other way compared to regular phantoms: they are spirits of the not-born-yet rather than spirits of the dead...

Amar Hadzihasanovic (Dec 07 2020 at 13:03):

I suppose pure motives are another quintessential mathematical phantom.

Amar Hadzihasanovic (Dec 07 2020 at 13:04):

Or maybe that should be mixed motives. I don't really know much about it.

Fabrizio Genovese (Dec 07 2020 at 13:17):

Matteo Capucci said:

In fact, I'd say pure phantoms do not last long since once you realize you may want them you're halfway through fomalizing them. A different kettle of fish is to formalize them right. E.g., $\Bbb F_1$ has been formalized for at least 25 years but we are still in the pre-paradigmatic phase of the associated Khunian revolution.
Infinitesimals are probably stuck there too.

I wouldn't say so. Infinitesimals are heavily formalized and a stable field by now. Non-standard analysis has consolidated a lot, it's sythetic differential geometry that tries to embrace a new perspective on the matter. Personally I'm not a fan of it. :smile:

Jens Hemelaer (Dec 07 2020 at 13:40):

Regarding the uniform probability distribution on the real line, I think the following paradox is interesting (I don't know if it has a name):
There are two envelopes with money, one with $x$ dollars and one with $2x$ dollars, where $x$ is distributed uniformly over the positive real numbers. You get one of the envelopes, and after opening it, you get the chance to switch to the other envelope, if you want. To maximalize the expected amount that you get, should you switch envelopes or not?

Shea Levy (Dec 07 2020 at 13:44):

Switch if your envelope is less than $\infty \over 2$

Jules Hedges (Dec 07 2020 at 14:15):

A general class of examples is "things that exist in a conservative extension of your theory"

Jules Hedges (Dec 07 2020 at 14:16):

For example if you're working in ZFC then you can pretty much get away with pretending that proper classes exist most of the time, even though plain ZFC doesn't think they exist

Morgan Rogers (he/him) (Dec 07 2020 at 15:05):

Or classifying objects which aren't representable (things which exist in eg a topos associated to your category, but don't pull back to concrete things in the category)

Fawzi Hreiki (Dec 07 2020 at 15:19):

Yeah I would say infinitesimals are basically dealt with. That’s not to say we fully understand them, but the direction of the axiomatisation is well worked out

Fawzi Hreiki (Dec 07 2020 at 15:21):

Fabrizio Genovese said:

Matteo Capucci said:

In fact, I'd say pure phantoms do not last long since once you realize you may want them you're halfway through fomalizing them. A different kettle of fish is to formalize them right. E.g., $\Bbb F_1$ has been formalized for at least 25 years but we are still in the pre-paradigmatic phase of the associated Khunian revolution.
Infinitesimals are probably stuck there too.

I wouldn't say so. Infinitesimals are heavily formalized and a stable field by now. Non-standard analysis has consolidated a lot, it's sythetic differential geometry that tries to embrace a new perspective on the matter. Personally I'm not a fan of it. :)

I'd be interested to hear why you aren't a fan actually. The first time I came across SDG I was blown away by how simple everything became.

Fabrizio Genovese (Dec 07 2020 at 16:32):

Me and other people tried to use it in a paper about categorical quantum field theory and things didn't really work out as expected. Then we tried with non-standard analysis and everythin went really smooth. Just a matter of bad experiences I guess?

John Baez (Dec 07 2020 at 17:46):

Amar Hadzihasanovic said:

I suppose pure motives are another quintessential mathematical phantom.

Motives are a great example of a phantom that Grothendieck was chasing. There are various ways to formalize them, but the hard part is getting one that one can use to prove all the conjectures about them.

Very very crudely, you want to take the category of projective algebraic varieties and freely turn it into a semisimple abelian category, so every algebraic variety is a direct sum of "building blocks".

Peter Arndt (Dec 07 2020 at 18:13):

The machinery of topos theoretic spectra is a midwife helping unborn spirits into existence: E.g. the initial local ring under a given ring in general doesn't exist ... in the category of sets -- but it does exist in another topos!

Matteo Capucci (he/him) (Dec 08 2020 at 14:16):

Peter Arndt said:

The machinery of topos theoretic spectra is a midwife helping unborn spirits into existence: E.g. the initial local ring under a given ring in general doesn't exist ... in the category of sets -- but it does exist in another topos!

This a beautiful and eerie image at the same time

Reid Barton (Dec 09 2020 at 12:33):

In homotopy theory there is a phenomenon that if you look at (say) the homotopy groups of spheres at a fixed prime $p$, there are certain patterns where nothing appears until degree roughly $p$, then the next term is in degree roughly $2p$, and in general there is a simple description up to degree around $p^2$, and then a more complicated description until around $p^3$, and so on. But for any fixed prime $p$, this description breaks down at some point because the powers of $p$ don't grow quickly enough.

Reid Barton (Dec 09 2020 at 12:34):

So there's this funny idea of a hypothetical "infinite prime" (due I think to Haynes Miller?) at which all of these components would be spread out infinitely far from each other, and could never interfere.

Reid Barton (Dec 09 2020 at 12:34):

I'm not familiar enough with this area to know whether the "infinite prime" is really a "phantom" in the sense discussed here, though, or some other kind of hypothetical object.

Matteo Capucci (he/him) (Dec 09 2020 at 14:11):

Very cool! Indeed, isn't number theory also very interested in the infinite prime, namely the characteristic of "characteristic 0" fields?

Reid Barton (Dec 09 2020 at 14:14):

I think, but am not sure, that this infinite prime in topology is a quite different thing.

Ben MacAdam (Dec 09 2020 at 15:22):

Fabrizio Genovese said:

Me and other people tried to use it in a paper about categorical quantum field theory and things didn't really work out as expected. Then we tried with non-standard analysis and everythin went really smooth. Just a matter of bad experiences I guess?

Just out of curiousity, were you working with general R-modules or modules satisfying the Kock-Lawvere axiom?

John Baez (Dec 09 2020 at 19:58):

Reid Barton said:

I think, but am not sure, that this infinite prime in topology is a quite different thing.

I think that's completely different - if they turned out to be the same it would be a big deal! The infinite prime in number theory is another phantom: see e.g.

• Shai M. J. Haran, The Mysteries of the Real Prime, Clarendon Press, Oxford, 2001.

Matteo Capucci (he/him) (Dec 09 2020 at 20:36):

Any intuition to share about why they should be different?

John Baez (Dec 09 2020 at 21:00):

Why should they be the same? In one case we're noticing a pattern for homotopy groups involving ordinary finite primes, noticing that the pattern gets easier to state in the limit $p \to \infty$, and start wishing there was an infinite prime so we could avoid talking about limits. In the other case we're noticing that all valuations on $\mathbb{Q}$ are of two kinds: the ordinary absolute value $|x|$ and the $p$-adic valuations $|x|_p$ coming from primes $p$. Then we start dreaming of an "infinite prime" or "real prime" so we can treat the ordinary absolute value like all the rest.

They just seem very different.

John Baez (Dec 09 2020 at 21:01):

The second story gets a bit fancier than what I said when you get into Arakelov theory: then there's a "real prime" and a "complex prime". But I don't think that matters much here!

John Baez (Dec 09 2020 at 21:18):

Regarding the second case: there's always been this battle between prime ideals and valuations in number theory, and this pressure to think of valuations (like the ordinary absolute value on rationals) as weird mutant prime ideals. Arakelov theory is the current attempt to settle this fight. Neukirch's book Algebraic Number Theory is a great attempt to explain Arakelov theory in a minimally terrifying way. I need to keep rereading it.

John Baez (Dec 09 2020 at 21:45):

Reid Barton said:

In homotopy theory there is a phenomenon that if you look at (say) the homotopy groups of spheres at a fixed prime $p$, there are certain patterns where nothing appears until degree roughly $p$, then the next term is in degree roughly $2p$, and in general there is a simple description up to degree around $p^2$, and then a more complicated description until around $p^3$, and so on. But for any fixed prime $p$, this description breaks down at some point because the powers of $p$ don't grow quickly enough.

I reblogged this on the n-Cafe and we now have some expert commentary. Interesting!

Simon Pepin Lehalleur (Dec 26 2020 at 11:41):

Maybe "the Shimura variety for GL_n/Q" is an example of a mathematical phantom?

Here is some context. Most reductive groups over number fields are not attached to Shimura varieties (ie the associated locally symmetric spaces are not algebraic); in particular, the group GL_n/Q does not have a Shimura variety for n>2. This is bad, because Shimura varieties are essentially the only known way to prove cases of the global Langlands correspondence, and the Langlands correspondence for GL_n is supposed to be the most basic one!

Nevertheless, people have found many interesting workarounds using Shimura varieties for other groups (in particular unitary groups), and in other situations (local and global function fields, and much more recently p-adic fields in the work of Scholze) there are algebro-geometric objects which play the role of Shimura varieties in their respective context in the sense that the appropriate Langlands correspondence is realized via their cohomology. So there is a faint hope that one day there will be a "Shimura variety for GL_n".