This peculiar thought experiment captures the first time I've been able to completely flip the idea of counting, with a rather original approach. Other attempts have been more high-level, with fewer details, and couldn't be pursued very far. But this time I got to the bottom. Please be patient with this idea. It gets messy at points, but I think the underlying restructuring can be done, even if only as a novelty.
Note, I do not think my proposal below is a sensible one. It is composed of pieces which don't yet fit together well -- and one of them is completely absurd -- but I'm leaving it in place because it is grasping at an intuitive hunch that I haven't been able to figure out, which may resolve it.
This is a step on a larger journey. I've been pondering this kind of inversion for years, in many small pieces, not knowing how to do it all together until I recently read a description of Gödel's construction of L, which begins with the empty set.
I do not like the empty set because it confuses me, feels like a paradox instead of something reasonable, so my mind began working on eliminating it from his structure while I read. I replaced "empty set" with "infinity" to see what happens, and suddenly, in a flash, I saw a way to make this happen.
First, I made a meme with the text I was reading.
Then as I thought more about the idea, I wrote the following:
Mathematics arises from division (which arises from naming things, which arises from language) and pattern recognition. The ancient Greek understanding of "1" makes this point more plain than what we understand today. For ancient Greeks, "1" was indivisible, and therefore not a number. Therefore the implied role of division as a core feature of mathematics was more obvious then than it is today, when we take it for granted and no longer see it hiding beneath everything.
Math started with counting things.
(I continued writing.) Most people imagine "math started with counting things," and intuit addition as the first operation, but look more closely at what happens when counting. We're overlooking an important step when we assume addition is the first thing that happens.
If you step back far enough in what is actually happening, you see counting begins by dividing the first item from everything else under the sun. After this separation happens, then the count is applied to this item. We call this item number "1." We know the "1" is an abstraction temporarily assigned to the real thing which itself is not "1," but is called "1." We do this again, and call the 2nd-divided thing "2." We repeat this for the third item, then the fourth...
This is how counting starts with division, not addition.
So here is a sketch of an idea for how to develop something structurally similar to naive set theory, except starting from the "big" end instead of the "empty" end where Gödel (and everyone else) started.
Our intent will be to develop mathematics without requiring an empty set, a pointless point, or any other such paradoxes or leaps of faith, while also being more clear about the act of division which underlies all counting.
Start with the set of uncountable infinity.
This is the sea out of which our math arises. We start here, at the other end of the nothing-to-everything spectrum from where we normally start. There's a reason. We're doing this instead of repeating the same mistake Euclid made when he created "points that don't exist," and which Gödel reiterated when he created L by starting with the empty set.
As you follow this narrative, your mind may resist starting with "uncountable infinity" because it wants to start with simple things and build complexity, but as you go through this thought experiment, you'll see that we actually are doing that, just in a different way than you may have ever imagined.
Then establish an operation called division.
Using this operation, extract the set of countable infinity from the set of uncountable infinity. Extract it one item at a time, though -- not all at once. At least, not yet. When we look at things from the big end, we see the smaller countable infinity arises, one number at a time, from the larger uncountable infinity, which makes a lot more intuitive sense than the other way around, if you think about it. Hopefully you see the value of drawing countable out of uncountable, because it's the last sensible thing we'll do for a little while.
We'll need to change the meaning of one.
Although we're going backwards from the way we normally think of things, we do not start with the infinite-end of countables using huge numbers, counting backwards, like 1,000,000,000,000,000, then 999,999,999,999,999... but we will use 1,2,3,4,5,6,7... just as normal counting goes. This is probably going to break your brain the first time you try it. If you're like me, you'll feel like you're turning your shirt inside out while running backwards up a hill toward a cliff's edge -- for no good reason. If it hurts, I suggest skimming the following text to see if you can get the overall gist before diving in to the weird details and quickly giving up.
Note that the meaning of "1" will need to change. It no longer means "the smallest number" in a context where numerical quantities increase with counting, but "1" is now "the largest number," referring to quantities which will be decreasing as we count. In other words, instead of "1" meaning something tiny, it means that thing that we normally mean when we say "bazillion," that is, an inconceivably large number at the end of all numbers, just before endless infinity begins. The number at the end of countable infinity. With this approach, the number one now lives at the border of countable and uncountable, the opposite from its normal meaning of smallness. It also, of course, is referring to something small, the thing you're counting. If it feels weird for "1" to have two contradictory meanings (big and small), note that we're not completely abandoning reason: our ordinary "1" also has both of those two meanings, but they are more harmonized: "conceptually small" and "referring to something small." (I have a suspicion this confusion is more an artifact of the way we normally think about numbers than something which is inherently confusing. Or it may be an artifact of how we're proceeding, and someone else may find a better way of saying this part. Or I may be just plain wrong, that's okay also.)
But wait, before we get too far into one, let's look at zero.
We're so used to beginning with one, we normally forget the role of zero until we specifically need to use it. But here, being explicit about everything we can, the first thing that gets "counted" is not what we are going to assign "1," but instead, the first thing that is counted is what we know as everything that existed before counting begins. This we assign to 0.
Computer programmers may find this step to be easy, but people who are not used to beginning counts with zero may find this awkward. In short, zero no longer represents "empty" as we are used to, but rather "everything."
Note that here, we're not just talking about everything mathematical, but everything, including yourself and your thoughts. The whole universe. There's a reason for including everything which we'll get to shortly.
This new idea of zero is roughly analogous to the normal idea of infinity... but way, way, bigger, because it encompasses everything, not just mathematical things. This idea really means to encompass E v e r y t h i n g, with a capital E and in bold. All of this together is now called zero. Yes, we're in effect saying zero = infinity. There is a little more going on than that though, and everything is about to go backwards from normal, so that's just a rough analogy to help get the idea across.
This is probably the hardest part to wrap your mind around. If normal infinity is an abstract, mathematical, flowerpot, this zero represent an actual flowerpot with many flowers growing in it, only one of which is the previous idea of infinity.
Within this approach, when we say that something is countable as one we are implicitly saying zero plus one meaning everything plus this new assignment of countability as the number "one" to the one thing we are counting. This thing we are counting existed before counting, and continues to exist independent of being counted.
Now we set the foundation and we're properly ready for our one. Because zero is "everything that existed before counting" we are therefore "drawing one" out of "zero," instead of "adding one" to nothing. As described earlier, this one is conceptually enormous, being nearly as big as zero, which is everything. One represents "the next smaller thing than everything" even though it is being assigned as a count to one small part of everything. Once you get familiar with the way this is happening, you can ignore the underlying size, meaning you can count like you're already used to, but we're being explicit about hidden steps right now.
Finally, we are ready for two.
At this point in the experiment, when we say that something is "two," or "2nd," we are now implicitly saying "zero plus one plus another one." In this way, zero -- everything -- is silently included in every mathematical statement. There is no longer an unmentioned separation between math and physics; they meet at zero, which is referring to "everything," which includes all matter, not just abstract math as we're used to. We're reiterating this point for the third time because it is so counterintuitive, hopefully it is somewhat clear now.
Now we are ready to build the rest of mathematics.
We have turned math entirely on its head -- or inside-out -- and set it upon a solid foundation which involves physics, instead of upon a paradox. What does the rest of mathematics look like as we cascade this new way of seeing things outward through the rest of the landscape?
To summarize our perspective, we're counting backwards from an uncountably infinite mathematical zero, to a penultimately* enormous one, to an enormous but slightly smaller two, to a smaller but still enormous three, and so on, toward an eventually infinitismally small quantity which is labelled with a very large number, something like "sesqui-milli-mega-ptrillion," if you will. The name is not important, just like the name of any enormous quantity in normal math is not something we care about until we (rarely) get to it.
Rather than a large number referring to a very large physical size as we're typically used to, this very large number is referring to a very small size in physics, ultimately arriving at Planck smallness, although theoretically we could go smaller if we needed. Conceptually, it may help to see this happens in much the same way that 1/10 refers to something smaller than 10... only we haven't introduced fractions into our study yet. If it feels like we're working with fractions, note that, in fact, we are, because we started with division, but also, we're not, because we haven't discussed the particular sort of division (a division of a division) that produces fractions, much less reals (a division of a division of a division). We're still just working with positive integers at this point, in order to lay a solid foundation.
What is clear: lots of things change in this rather original approach to mathematics, but from all appearances much of "normal" mathematics can be preserved. For example, without investigating it yet, I wonder how it might affect calculus, possibly straightening it out right at the point where it discards infinitismals -- which we can now talk about without that nagging feeling that we've disposed of something which could possibly be important someday.
This is a rough sketch. The underlying logic in a few places is loose and may be confusing without illustrations that break each step down. There are potential paradoxes which may end up making it completely useless, or may be able to be resolved by others. I actually tried making illustrations but soon decided I am not comfortable with doing that yet, since some of these pieces may require tinkering and reorganization before visuals get created. I have not properly explained why I emphasize that 1,2,3... represent very large concepts while simultaneously referring to small things, but I don't know how to put that part into words yet. I also didn't expect the thought experiment would end with two infinities (I was trying to fix that) and no way to handle negative numbers yet -- all of which makes me nervous --, but I think the overall structure holds merit. [A later edit and I'm starting to see a way to handle negative numbers, and the dual infinity is not really as much of a bug as it is in normal math because one arises from the other and is quite different from the first -- instead of them equally opposing each other -- but that's a story for another day]. I feel like there is enough here for others to begin exploring the approach -- or at least tell me that someone like Euler, Gauss, Poincare, Ramanujan, or Gödel already did this once and nobody cares. There is something underlying the "new meaning of one" which I couldn't capture in words yet. [Another later edit and I am wondering what if we imagine we're counting everything from its inside, not from its outside, meaning that our enormous one that represents nearly-everything is representing every thing from within each thing, if that makes any sense]. I have a hunch that the division property we normally associate with "1/2," famous from Riemann's insight into primes, plays a curious role in the not-entirely-clear transition between zero and two, but I don't know how to articulate it yet. In the end, perhaps what saves this essay is that I wrote it on April 1, although to be honest, I meant it sincerely and see it as an important step on a long journey. To be continued...
[Update, much later again. Today, April 7, 2024, I was watching a Veritasium video on p-adics and finally discovered the heart of why they're so useful. It's this: The prime-adics handle zero differently than base-10-adics, in a way that makes them more useful. This is nice to finally know, but I also discovered something much more relevant to this present weblog post. I'd seen it before but never made the connection to this present article, which I wrote before I discovered p-adics, otherwise I probably would have seen the alignment. The weird structures of this article are admittedly fragile and experimental, but some of their core conceptual structures, their inside-outness, align very well with p-adics, most particularly how the p-adics handle extremely huge numbers, like 1 less than infinity, and so forth. They align so well that I'm almost ready to drop this old thought experiment and take up the formative ideas here within the p-adic system instead. This is very exciting, and I already want to do something slightly different with ...999999.0 than what is normally done with it...]