Two Ways to Truth

If you have seen a hundred white swans, do you know with certainty that all swans are white? No. You know only one thing: that, so far, you have seen a hundred white swans. One black swan is enough to overturn a conclusion that seemed self-evident for generations. This matters because most of what we call “knowledge” works in exactly this way: we see, measure, experience, repeat the same observation often enough – and then infer a rule.

This form of reasoning is called induction – reasoning from particulars to a general rule. The broader approach that rests on it, according to which all knowledge begins in experience – in the senses, observation, measurement, and the world as it appears to us – is called empiricism. It is the most natural and intuitive way to think about knowledge. Want to know whether it is raining? Look out the window. Want to know whether a medicine works? Test it in a controlled trial against a control group.

Yet for all its power, it has one limit that cannot be avoided: it can never reach certainty. Induction can be very strong and support a conclusion to varying degrees – but it never proves it. It gives us probability and always remains open to revision. Notice the small word hidden inside every such conclusion: “probably.” Probably all swans are white. Probably the sun will rise tomorrow. Many in science see this as the source of its strength – it corrects itself in light of new evidence. But that same flexibility is also its limit: it never gives us fixed truth, only the best description we currently have.

Opposite it stands another path, one that begins not with the eye but with reason. It asks: are there things we know to be true – not because we have seen them, but because they must be true?

Premise A: All humans are mortal.
Premise B: Socrates is human.
Conclusion: Therefore, Socrates is mortal.

There is no need to test Socrates in a laboratory. If the two premises are true, the conclusion follows necessarily – not as something “likely,” but as something unavoidable. It does not depend on the weather, culture, mood, or angle of view. Or take an even simpler claim: a bachelor is unmarried. There is no need to survey a million bachelors. The concept “bachelor” is already defined as someone who is unmarried. Anyone who says “a married bachelor” contradicts the meaning of the word.

This form of reasoning is called deduction – reasoning in which the conclusion follows necessarily from the premises. The approach that sees reason, rather than the eye, as the source of certain knowledge is called rationalism.

The attentive reader will notice that the argument about Socrates can be stripped down to a logical structure that can also be written mathematically:

Premise A: Every X is Y.
Premise B: Every Y is Z.
Conclusion: Therefore, every X is Z.

This brings us to deduction’s strongest domain: mathematics. To know that 1+1=2, you do not need to place two apples beside each other. Apples may help a child understand, but they are not the source of the truth. 1+1=2 was true before the apple, after the apple, and even in a world without apples. It does not wait for observation, and no laboratory experiment can disprove it.

And if the senses are not a transparent window onto reality but a biological translation of it, a question opens up: is there any way to reach what is not merely a translation? Induction cannot take us there. It is confined to the realm of observation and can say only: “This is how things have appeared so far.” Deduction goes beyond it. It says: “This is how things must be.”

Two paradigms now emerge, and each comes at a cost.

Empiricism is deeply tied to the world around us. It measures, touches, repeats, and tests – which gives it immense power. But it buys all this at one fixed price: it is never certain. Everything it knows, it knows only “so far.” A black swan may arrive tomorrow. That is a serious price to pay if truth – and not merely usefulness – is the aim of inquiry.

Deduction, on which rationalism rests, makes certainty possible. 1+1=2 will not wait for the next study, nor retreat before any observation. It holds truths that do not change, leaving no room for surprises or probabilities: this is truth with 100% certainty, independent of time, place, sensory experience, or subjective perception. No “new knowledge” and no observation can overturn it, because it follows from pure logic itself – from the definitions of the concepts and from what must follow from them.

But deduction also has a cost, and it is precisely the opposite: it seems detached from the world. How could logic or pure mathematics grasp the smell of morning coffee, the weight of a stone in the hand, or rain on the skin? It seems to purchase certainty at the price of detachment from living reality.

This is the crossroads. One path offers contact with the world without certainty. The other offers certainty that seems detached from everyday life. Modern scientific culture tends, rightly from its own perspective, to prefer the first path: what can be measured, observed, and tested. But before deciding, it is worth pausing over that word: “seems.” If the second path – the one that appears detached – turns out to be closer to what truly exists, the whole choice is reversed. The challenge for rationalism is to show how it describes the world we live in, rather than merely an abstract one.

A common objection to rationalism, logic, and mathematics is the thought that perhaps all this stability is an illusion. Perhaps we defined the concepts ourselves, and if logic and mathematics are merely a language we invented, then they contain no truth independent of us – only agreements we have made with ourselves.

So let us be precise. We do determine the definitions. We do not determine their consequences. We defined a “prime number” as a natural number divisible only by itself and by 1. But once that definition exists, it was not we who decided that there are infinitely many prime numbers. We discovered that. No committee voted for it, no culture chose it, and no mind invented the necessity. Euclid proved it 2,300 years ago, and he had no choice in the matter. The proof is open to anyone willing to follow it.

This is the difference between invention and discovery. The symbols with which we write mathematics are inventions – it makes no difference whether we represent “two” as 2, II, or any other arbitrary mark. But the logical relations those symbols reveal are not subject to our will. The Pythagorean theorem was true before anyone formulated it, and it will remain true after the last human being disappears. The logic behind it will remain the same even on an alien planet. You can change its language; you cannot change what follows from it.

So far, we have been dealing with logic. But the same thread appears when we return to the senses. When we see a color, hear a sound, smell an odor, or feel a texture, we do not encounter the world “naked.” We do not encounter “the thing itself,” but always encounter it through a sensory system that filters and translates. Yet every translation requires a source. There can be no translation without something being translated.

Take the red of a flower. It is not “out there” as you experience it. It arises in the encounter between light, eye, and brain. Yet something out there does not depend on your eye – some structure of reality that the human eye translates as red, another eye translates differently, and a measuring instrument describes in wavelengths. Three different translators, one source. And that is precisely the point: wherever there is translation, there must be something being translated.

What we are looking for, then, is not experience as it appears to us, but what stands behind it. Not color, but what makes color possible. Not sound, but what makes sound possible. The world as it is before it has passed through any sensory system. Put simply: if the senses give us the translation – what is the source?

And that source, if it is truly a source, cannot depend on the eye that sees it. It must be what remains constant even when the translator changes: human, bee, bat, dog, measuring instrument – even an alien. That which does not change with the observer, we will call truth. Not “my truth” or “your truth.” Not truth as a feeling, a position, or a narrative. Truth in the strong sense: what remains true even without anyone to experience it, agree with it, or give it a name.

Let us return for a moment to the charge against rationalism: that it is detached from the world, playing with abstract ideas while science touches the things themselves. Look again. Every observation, every measurement, every experiment presupposes an order: that numbers will behave as expected, that logic is valid, and that what led to a conclusion yesterday will lead to one again tomorrow. The scientist who measures does not prove these assumptions – he relies on them before he has even touched an instrument. Empiricism does not stand alongside rationalism as an equal. It stands upon it. Before every induction, there is already a truth that no one has measured.

This asymmetry appears in the history of each. Nietzsche put it sharply: “Only that which has no history can be defined.”¹ Science has a history – a long chain of corrections. The geocentric model gave way to the heliocentric one, Newton was replaced by Einstein, and even relativity, physicists know, is not the final word. Mathematics has no such history. Mathematics has a history of discoveries, not a history of corrections to truth. The Pythagorean theorem has not been “improved” or “refuted” in 2,500 years, because there is nothing in it to correct. Once its axioms are laid down, it follows from them necessarily.

What seemed detached from reality begins to look like something placed at its foundation. Color changes from eye to eye, sound from ear to ear – but the structure beneath them, the order that allows every translator to translate, does not move.

This does not solve the question of reality, nor does it say that the world is “made” of mathematics – that will require another step, and not today. But the question deepens on its own. Not only how we know a truth independent of us, but where such truth exists at all. Not in the mind alone, because minds are born and die. Not on paper alone, because paper burns. Not in our agreement, because even an entire world, unanimously, cannot vote that 1+1=3. Whether we agree or not, numbers do not wait for our approval.

Experience depends on the one who experiences it. Logical truth does not depend on the one who thinks about it. And once something does not depend on us, it stops looking like a convenient game of symbols and begins to look like a thing in its own right.

Is the independent source behind the senses the same thing as logical truth, which is independent of us? That is precisely the question. But if it is – if the same order presupposed by every measurement and every inference is also what stands behind the translation – then perhaps we are not describing reality from the outside through it. Perhaps we are encountering that through which reality has form in the first place.