Wednesday, February 25, 2026
Elder of Ziyon
My position is that truth is real but that humans cannot access it completely through reason alone, though we can approach it. In other words, I am trying to prove that we cannot prove anything.
Yes, I hear the circularity. Stay with me.
The most obviously true statement I can think of is 1+1=2. Everyone agrees on it. Children learn it before they can read. It is the foundation of every calculation ever made. If anything is provably true, surely this is.
But is it?
Within mathematics, arithmetic rests on a foundation of axioms, starting points that are simply asserted rather than derived from anything more fundamental. Every system has to start somewhere, and mathematics starts here. Within that axiomatic system, 1+1=2 can be proven with perfect rigor. But the axioms themselves are not proven. They are chosen because they are useful and because the system they generate is internally consistent. The proof of 1+1=2 is only as solid as the unprovable assertions underneath it.
It gets stranger. Even within mathematics, 1+1 does not always equal 2. In arithmetic modulo 2, where numbers wrap around when reaching 2, 1+1=0. This is not a curiosity: it is the mathematical foundation of every computer ever built. It is a perfectly consistent system that describes real phenomena accurately.
And if you imagine a planet where two objects placed together always produce a third, the mathematics of that planet is not as fanciful as it sounds. Abstract algebra, a standard branch of mathematics, gives us a completely rigorous framework called an isomorphic field where you can create a system where 1+1=3. Define addition as a⊕b = a+b+1 and adjust multiplication accordingly, and every rule of arithmetic you learned in school still holds perfectly: commutativity, associativity, distributivity, all of it. The system is internally coherent in exactly the same sense that standard arithmetic is. The only surprise is that in this system the conceptual zero, the number that leaves everything unchanged when you add it, turns out to be -1 rather than 0. The whole structure shifts, consistently, and keeps working.
This is not a mathematical party trick. It demonstrates something fundamental: consistency does not pick out a unique truth. You can build multiple, mutually inconsistent arithmetic systems that are each internally valid. Standard arithmetic is not the one true math because it is the only consistent option. It is the one we use because it maps most conveniently onto the physical world as we ordinarily experience it. The choice was always pragmatic, not absolute.
But, you might reasonably say, mathematics is abstract. Out here in the physical world, if I add one marble to another marble I have two marbles. That is not an axiom. That is just what I can see with my own eyes.
Is it?
Consider what we are actually assuming when we count two marbles. We are assuming that a marble is a discrete object with a stable identity, that it neither combines with nor subdivides from other objects when we are not watching, that “adding” means placing in proximity without any interaction that changes the objects, and that we are counting objects rather than, say, colors or masses. These are all reasonable assumptions. They are also all hidden.
Start making them visible and the arithmetic gets complicated fast.
One blue marble plus one red marble is two marbles. But it is also still one blue and one red. Which answer you give depends on what property you decided to count before you started.
One marble plus one car is two objects. But you are sitting in the car, and the car is sitting on the road, and the road is resting on the earth. If we are counting objects in the scene, why do the car and the marble count but not the road? Because you drew a boundary around the system before you started counting, and that boundary is a choice, not a discovery.
One marble plus one car with your adorable five-year-old as a recent passenger is two objects plus however many marbles are wedged between the cushions, plus the candies he stashed in the cupholder, plus the air freshener, plus the brake lights. If we are counting all the objects in the system, where does the system end? When you said “one marble plus one car equals two,” you had already smuggled in a decision about which objects count. The arithmetic came after the philosophy, not before it.
One water droplet plus one water droplet is one larger water droplet. The arithmetic simply does not apply because the objects do not maintain their identity through the operation. The hidden assumption that objects stay discrete was doing all the work.
And one rabbit plus one rabbit, given a few months and a suitable habitat, may be much more than two rabbits. The arithmetic was never wrong, exactly. It just pretended that objects are static snapshots rather than ongoing processes, which real physical objects never actually are. Every object is a river, not a rock, and 1+1=2 works by freezing the river long enough to count it. One biodegradable cup plus one Styrofoam cup plus time equals one cup.
In every case, the apparent certainty of 1+1=2 dissolved when we examined the assumptions holding it in place. None of those assumptions are unreasonable. Most of them are exactly right for most purposes. But they are assumptions, not foundations. The proof was always conditional on choices we made before we started counting.
And if this is true of 1+1=2, the most obvious truth imaginable, it is surely true of every other claim anyone has ever made.
This is not itself proof that nothing is provable, or I would be contradicting my own argument. But it is a strong indication that absolute truth is normally inaccessible to humans through reason alone. We reason within frameworks. Every framework rests on axioms. No axiom is self-proving. The turtles do not go all the way down. At some point there is just water.
This should be paralyzing but it is not, because we already know how to live with it.
1+1=2 is true enough to build space stations. Newtonian physics is false in the sense that Einstein corrected it, but it is true enough to design train schedules without worrying about relativistic effects. The question is never whether a framework gives you absolute certainty. The question is whether it gives you adequate accuracy for the domain and the stakes involved. We build on good-enough foundations because we have no other kind.
This matters for a lot more than mathematics. A significant strand of Western philosophy, going back to the Greeks, is built on the assumption that human reason can access complete truth directly, that by thinking hard enough and carefully enough we can arrive at foundations that need no further support. If that assumption is wrong, then everything built on it needs reexamination. Our epistemology, our ethics, our political theory, our institutional design: all of it looks different once you accept that the foundations are chosen rather than discovered, adequate rather than certain.
Here is the consequence I find most interesting, and the one I will develop in the next piece.
If we cannot know absolute truth, then the trivial definition of falsehood as “not truth” becomes unintelligible. We do not have a fixed point to measure distance from. This means we need a separate epistemology of falsehood, a way of identifying and eliminating the false that does not depend on first establishing the true.
It turns out that while truth is not provable, falsehood often is. And building that epistemology of falsehood is something I have been doing, without quite realizing it, for the past two decades, in the very different context of debunking lies about Israel. The tools turn out to be the same.
More on that next time.
|
"He's an Anti-Zionist Too!" cartoon book (December 2024) PROTOCOLS: Exposing Modern Antisemitism (February 2022) |
 |
Buy
