Could God Make 2+2=5? Descartes and the Puzzle of Necessary Truths
The first-meditation trick: clearing your mind with falsehoods
In 1641, René Descartes (1596–1650) published his Meditations on First Philosophy. He did something that still startles readers: he openly wrote claims he knew were false. In the First Meditation, he let his imaginary thinker wonder whether a supremely powerful God might make us go wrong every time we add two and three or count the sides of a square. He even floated the idea of an evil demon who constantly tricks us about everything.
But Descartes was not really being reckless. He was using a teaching strategy, what he called the analytic method. At the start of inquiry, most people hold confused ideas — we think we know things through the senses, or that colors exist in objects the way they seem. Descartes believed that if he just stated the truth, readers would hear it through those confused ideas and misunderstand it. So he met them where they were. He wrote in the voice of a meditator who slowly discovers that what seemed possible at first is not possible at all.
This matters because Descartes had a strict rule for doing metaphysics: only trust what you clearly and distinctly perceive. A perception that is clear and distinct is one that is so sharp and obvious your mind cannot deny it — like seeing that a triangle has three sides. In contrast, many of the possibilities his meditator entertains are not clear and distinct; they are leftover muddles from everyday thinking. Descartes later judged that the idea of God as a deceiver is not a real possibility — it is a confused notion that disappears once we examine what God truly is. So the radical doubts of the First Meditation are no part of his actual philosophy. They are props for clearing your mind.
Eternal truths: did God invent math?
Descartes’s most famous claim about necessity is that eternal truths — propositions like “2+2=4” or “all radii of a circle are equal” — were created by God. He wrote to his friend Marin Mersenne in 1630 that God established these truths “by the same kind of causality as he created all things, that is to say, as their efficient and total cause.” This flips our normal intuition upside down. Usually we think mathematical truths could not have been otherwise; they are necessary, not just true by chance.
But Descartes insisted that God was perfectly free to make the radii of a circle unequal, just as free as He was not to create the world. That sounds like God could have made 2+2 equal 5, or made contradictories true together. Yet Descartes also said that the necessity of these truths “does not surpass our knowledge” — we can be certain of them. So right away we face a puzzle: if a truth is necessary, it seems it could not have been different. But if God created it by a free act, couldn’t He have created a different truth instead?
Descartes’s answer is that God’s freedom is not like ours. He wrote that God’s will “was not indifferent from eternity with respect to everything that has happened or will ever happen” — meaning God’s will is completely independent and not pushed around by anything outside Himself. This is what Descartes calls divine indifference. And crucially, for Descartes, God is immutable. His will is a single, simple, unchanging act. So while we might say, “God could have done otherwise,” Descartes’s picture of God as eternal and unchangeable makes that phrase slippery. If God never changes, and His will is one act for all eternity, then in what sense could an alternative version of “2+2=4” ever have existed?
Three ways philosophers tried to untangle the knot
Modern interpreters have wrestled with this tension. Harry Frankfurt took Descartes at his word: God really could have made contradictories true. If that’s so, then eternal truths are not truly necessary — they are contingent, because they could have been false. The feeling that they must be true is just how our minds happen to be built, not a reflection of how reality absolutely is. A problem with this reading is that if even the most obvious truths are only apparently necessary, it’s hard to see how we can know anything at all, including the claim that God is omnipotent.
Edwin Curley offered a more complex picture. He suggested that Descartes thought in terms of iterated modalities — possibilities about possibilities. God willed that certain truths be necessary, but God could have willed the opposite possibility (that those truths not be necessary). So a truth like 2+2=4 is necessary — it cannot be false — yet it was not necessarily necessary. Critics objected that if something could have been non-necessary, then its necessity is not absolute. And the idea seemed to limit God’s power: God could not make a necessary truth false without first un-making its necessity, like having to unlock a door before opening it.
Lilli Alanen emphasized that Descartes thought we simply cannot understand how God’s free will produces necessary truths. There is no modality — no “could have been otherwise” — that applies to God’s own activity, because God is the source of all possibility and necessity. The truths are fully necessary, but God’s way of authoring them is something our finite minds cannot grasp. This protects the necessity of math, but some philosophers worry that it dodges the problem rather than solving it.
In the background lies a further, bolder option: perhaps Descartes was a necessitarian. Some passages suggest that because God’s will is eternal and immutable, there never was any real alternative to the truths and events He decreed. That would mean all truths are necessary, and nothing could possibly have been different. Descartes himself seems to have flinched at this thought, but in a recorded conversation he reportedly said that God’s decrees “were also completely necessary” — though we should note that this comes from an interview, not a work Descartes himself published.
What this means for you: freedom and falling dominoes
Why does a 17th-century puzzle about God and math matter to you? Because it touches a question that still stings: are your choices free, or was everything you’ll ever do already settled long before you were born?
Descartes insisted that God preordains all things “from eternity.” He told Princess Elisabeth that “the slightest thought could not enter a person’s mind without God’s willing, and having willed from all eternity, that it should so enter.” If that’s true, then when you pick chocolate ice cream over vanilla, it might feel like a free decision, but behind that feeling lies an unbreakable chain that was always going to lead to that exact moment. Descartes tried to comfort her by saying that our experience of independence is not incompatible with a deeper kind of dependence on God. But calling this a comfort may not make it feel satisfying.
Many philosophers today still argue about whether the world is deterministic — whether every event is the inevitable result of prior causes, with no room for alternative possibilities. Descartes’s struggle with eternal truths is an early version of that worry. If the deepest truths of mathematics could not have been different, then what about the smaller truths of your own life? Can you really have chosen otherwise, or is that just a feeling generated by a mind that cannot see the whole chain?
Descartes never gave a final answer that made everyone nod in agreement. He appealed to God’s incomprehensible power: we can’t understand how freedom and preordination fit together. But his honesty about the difficulty set the terms for centuries of debate. The next time you think, “I could have done that differently,” you are stepping into Descartes’s very same puzzle.
Think about it
- If you knew a supercomputer could predict every choice you’ll ever make with perfect accuracy, would you still believe you are free? Why or why not?
- Suppose God had made 2+2=5 instead. Would that change what you count as a “good reason” for believing something? Could you ever know that it had been different?
- When you blame someone for a wrong action, you usually assume they could have done otherwise. If the world turned out to be completely determined, would blame still make sense?
