QuestionsCan we be certain of anything?The book of the universe is written in the language of mathematics
Can we be certain of anything?Yes, through reason

The book of the universe is written in the language of mathematics

The book of nature is written in mathematics

Galileo Galilei 1564 · 1642Astronomer and physicist: he holds that nature is read through experiment and mathematics, and that Scripture speaks of salvation, not of the mechanics of the heavens.
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What it claims

On what is mathematical in nature, the human mind can reach a certainty equal to God's, even if it knows very little of it.

Why

The universe is written in mathematical figures, and what we demonstrate there is known as surely as God knows it.

The same idea through history
Pythagoras450 BCThe Pythagoreans already saw in numbers what the world is made of. Galileo does not say that things are numbers: he says that their laws are written in triangles, circles and ratios.
Galileo Galilei1623In The Assayer, Galileo answers 'Sarsi' (the Jesuit Orazio Grassi), in whom he sees the belief that in philosophy one must lean on a famous author, as if philosophy were a work of fancy like the Iliad. No, says Galileo: philosophy is written in the book of the universe, in the language of mathematics.
Galileo Galilei1632In the Dialogue he says what this language is worth for our mind: on the mathematical propositions we understand, our certainty equals God's.
Albert Einstein1936In the 20th century, Einstein marvels that the world lets itself be grasped by our concepts. Galileo asserts that the world is mathematical; Einstein asks why that works.
In short

"The book of the universe is written in the language of mathematics" is an argument by Galileo Galilei (The Assayer, 1623 (§ 6); Dialogue Concerning the Two Chief World Systems, 1632 (First Day), 1623) on the question "Can we be certain of anything?" (answer: yes, through reason). Its claim: On what is mathematical in nature, the human mind can reach a certainty equal to God's, even if it knows very little of it. According to AskPhilosophy's logic check (Z3 solver), the argument is valid only if one grants a premise the author leaves unstated (P4). AskPhilosophy presents 3 objections, 3 of them with a documented reply, and the original text behind each premise, checked word for word.

The Assayer, 1623 (§ 6); Dialogue Concerning the Two Chief World Systems, 1632 (First Day)
Valid with an implicit premise

In 1623 Galileo answers 'Sarsi', pen name of the Jesuit Orazio Grassi, who had defended following the astronomer Tycho Brahe: whom else should one follow? Galileo sees in this the belief that in philosophy one must lean on a famous author. He answers that philosophy is not a work of fancy like the Iliad, where truth hardly matters. It is written in another book, always open before our eyes: the universe. That book is written in the language of mathematics, with triangles, circles and other figures. In the Dialogue (1632) he goes further: for the propositions of geometry and arithmetic that we understand, our certainty equals God's, even if God knows infinitely more than we do. This idea became a motto of modern physics. It also raises a question nobody has closed: what does this certainty guarantee when we speak of the real world?

P1The universe is like a book written in the language of mathematics: its characters are triangles, circles and other geometrical figures. Without them, it is humanly impossible to understand a word of it.
P2Understanding can be taken in two ways. By the number of things understood (extensively), our mind is almost nothing, for truths are infinite. By the perfection with which a proposition is understood (intensively), the human mind understands some of them (pure geometry and arithmetic) as perfectly as nature itself.
P3For these propositions, human knowledge equals divine knowledge in objective certainty, because it grasps their necessity, and no greater security seems possible.
P4 · implicit premiseWhat is demonstrated in mathematics applies to the book of the universe: reading nature with these figures and ratios means reading propositions we understand with that certainty.
Therefore, on what is mathematical in nature, the human mind can reach a certainty equal to God's, even if it knows very little of it.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.
Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Without the implicit premise: Without P4, the computer finds a case where nature is written in mathematics, where we have perfect certainty about demonstrated mathematical propositions, and where we nevertheless have no certainty about nature: the demonstrated propositions may speak of ideal figures, not of real bodies.
The argument is valid with the hidden premise P4. P1 says nature is written in mathematics; P2 and P3 say that some mathematical propositions are understood with perfect certainty. To conclude to certainty about nature, one must add that these understood propositions are the ones that describe the world (P4). All the objections start there: a mathematician is certain of his theorems, but what does that prove about a planet or a falling body? Galileo also says this certainty is very limited: a thousand propositions, set against infinity, are like zero.
Analogy. You are learning a foreign language. Its grammar rules are sure: you know them like the best grammarian. But the book you read is immense, and you have read only a few lines. Galileo says: nature is that book, mathematics is its grammar. It remains to be seen whether the grammar you hold is really that book's.

Objections

3

Each objection is a conversation: click to see who replies and where the debate stands.

S
Simplicio (in Galileo's Dialogue) 1632 · attacks premise 3

Simplicio, Aristotle's defender, first sees a contradiction: human understanding is praised as nature's greatest gift, then said, with Socrates, to understand nothing. When Salviati answers by crediting it with a certainty equal to God's, Simplicio finds this way of speaking "very resolute and bold".

a defender replies
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Salviati (Galileo) defender of the argument · 1632 · Salviati's answer
He distinguishes the quantity and the quality of knowledge. In quantity, God knows infinitely more. But for the few propositions we do understand, the truth is the same: one cannot say that God can make what is done not done. Only the manner differs: we move step by step from conclusion to conclusion, God sees all at a single glance.
Where the debate stands (our summary)Our summary: Galileo's answer does not say man equals God, it says a theorem's truth is the same for everyone. That point has remained accepted, even by those who reject the rest. The discussion moved to the application to real things (see the following objections).
David Hume 1748 · attacks premise 4

The truths of mathematics concern relations between ideas. They are discovered by thought alone, and would remain true even if there were no circle or triangle in nature. Their certainty comes precisely from depending on nothing that exists. For facts about the world, the contrary is always possible, since it implies no contradiction: only experience can inform us, and it gives no demonstrative certainty.

Galilei replies
Immanuel Kant 1787 · Kant's answer
Kant accepts that mathematics is certain and refuses to reduce it to relations of ideas. Arithmetical propositions are always synthetic: they add to our concepts. Yet they hold for all possible experience, because space and time are the very forms of our experience. But this means that the certainty holds for the world as it appears to us, not for things in themselves.
see the argument
Where the debate stands (our summary)Our summary: Hume separates certainty (mathematics) from information about the world (experience). Kant tries to reunite them, at the price of limiting certainty to phenomena. Galileo asserts on the contrary that nature itself is written in mathematics; neither Hume nor Kant grants him that without reserve.
Albert Einstein 1921 · attacks premise 4

Einstein, a physicist who extended Galileo's idea, limits it himself: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Modern mathematics separates the formal (which is certain) from its intuitive content. When geometry is applied to the world, one makes a hypothesis that experience may refute.

Galilei replies
E
Eugene Wigner 1960 · The mystery that remains
Wigner does not answer Einstein by restoring Galileo's certainty. He notes a strange fact: the language of mathematics fits the laws of physics astonishingly well, and we do not understand why. He calls it a "miracle", "a wonderful gift which we neither understand nor deserve". Galileo's intuition would thus be confirmed by results, but without a known foundation.
Where the debate stands (our summary)Our summary: nearly everyone admits that mathematics succeeds in physics; they divide over the reason (nature is mathematical, the mind models it in its own way, or it is a happy accident). Absolute certainty about the world, however, is given up by modern physicists.