I’d like to talk about the skeptic argument, which attempts to show that true knowledge is impossible. I think that (a) the conclusion really shouldn’t be as crazy as we think and (b) the argument doesn’t rule out all types of knowledge.

First of all, let’s go through a standard version of the argument against knowledge. Here it is:

1. If you truly know something, then you must be able to rule out all possible doubt.
2. You can doubt everything, as your senses are unreliable.
3. Therefore, you cannot know anything at all.

The fact that we supposedly don’t know anything should not come as too much of a surprise. Take, for example, the idea of where we parked our car. Colloquially, we would say we know we parked our car in the lot, for instance. But at the end of the day, that’s just an assumption, and it shouldn’t take a skeptic to make you think that. Without proof your car is still there (or was there), you have no way of even beginning to justify the claim you know where your car is. I’d argue that the majority of cultural “knowledge” we have falls into this category: assumptions.

Whether or not you agree with my first point, I think the skeptic’s argument still hits a wall. The basic premise of the skeptic argument is that our senses are unreliable, i.e., they deceive us (optical illusions, for example). But I think it is a fallacy to lump all knowledge into empirical knowledge—things learned from the senses (trees, tables, other people). Knowledge that is a priori, or prior to sense, like math/logic, doesn’t depend on whether you are in a vat, deceived by an evil demon, or the only person in the world. For example, in a simple modus ponens argument, one can decree:

1. If Mary has a cat, the cat is red.
2. Mary has a cat.
3. Therefore, the cat is red.

Even if it is a demon deceiving you, the logic that holds these premises together still remains true. Sure, the demon could be deceiving you about everything else in the argument. Mary could have a dog in reality, and the dog could be blue. Regardless of that, the logical structure of A → B, B, therefore A, still remains unfalsifiable. As this set of axioms is internally consistent, it doesn’t matter if the soundness of the argument is questioned — it is valid in any scenario anyway.

Of course, here, skeptics bring up the classic argument that all logic — and mathematics — is based on axioms that are basically created inside of the mind. The conclusion of this rebuttal, from what I understand, is to undermine the validity of logical structures: if they are created by humans, and we are inherently flawed/subjective creatures, they cannot be trusted. One way to counter this argument is to simply say that math is intrinsically cemented in the universe, which (I think) seems closer to the truth. Another way, which Descartes himself used, is to decree that mathematical truths are dependent on God’s will. In Meditations, Descartes essentially argued an anti-Platonist view: that even the eternal truths of mathematics (like 2+2=4 or the sum of angles in a triangle being 180) are dependent on God. He argues that God freely created these truths, just as He created the physical world.

I’d argue that Descartes-esque responses are not really needed. Even if humans created the axioms to describe math, does that really mean that mathematical logic is an unreliable form of knowledge? I don’t think so, and I think this “gotcha” argument that mathematics is based on Euclidean geometry fails. First of all, there are many other systems of mathematics that are non-Euclidean, from what I understand: hyperbolic geometry, elliptic geometry, and many others. This just goes to show that multiple systems of mathematics exist, and that the entire discipline isn’t basically based on the thoughts and mental projections of one man.

Now, the next objection here is that the existence of these systems signifies that anyone can just up and create their own system. To this, I kind of agree—with the caveat that the premises must be internally consistent. This brings us to the real crux of mathematics in the modern age: the problem of consistency. It isn’t enough to simply invent a system of axioms; instead, the system needs to be internally coherent, or else contradictions can unravel the whole thing. This became clear after Russell’s Paradox demolished “naïve set theory,” showing that careless assumptions (like the idea that every property defines a set) lead straight to contradiction. The response was Zermelo–Fraenkel Set Theory with Choice (ZFC), which restricted what kinds of sets are allowed to exist. ZFC became the new foundation, as it avoids paradoxes and seemed consistent enough to build mathematics on. However, we can admit that ZFC is something human-made: simple criteria designed to prove that the mathematical and logical systems we use to describe our universe are consistent. What’s to say it won’t change in the future?

For now, as long as you can create a mathematical system that is internally consistent, i.e., in line with ZFC, it makes sense. But it isn’t just that simple: Gödel, in the 1930s, showed that no mathematical system is sufficiently strong enough to prove itself from the inside, meaning that it is possible ZFC contains some errors itself.

Despite this, we need to notice what this actually shows (i.e., the easier conclusion. I’m not giving a consistent argument for numbers outside the mind here, as I am a little conflicted on the thought): that the skeptic’s doubt cannot and does not annihilate logic itself. Even if some particular axioms collapse under paradox, the underlying logical structures — modus ponens, conjunction, modus tollens, and substitution—still stand. They’re the tools by which we discover those inconsistencies in the first place. The paradoxes in set theory are discovered through logical reasoning in the first place, which means logic is self-correcting, not self-destructive. In other words, skepticism about empirical knowledge, or even about mathematical systems, cannot overturn the validity of logical reasoning.

So while the skeptic might convince us that we can’t be sure our car is really in the parking lot, the fact remains: if A implies B, and A is true, then B follows. That kind of necessity is immune to demons, vats, or human subjectivity. Logic may not give us knowledge of the external world, but it does provide knowledge of something deeper: reason itself.

Lastly, this post would be amiss without mentioning the famous contradiction: the skeptic argument at the start of the post postulates logic as a valid form of knowledge—it itself utilizes a modus ponens argument, where two premises lead to a conclusion. And if skepticism relies on logic, doesn’t that mean that logic is outside the realm of doubt in the first place?