0 and 1 in Greek mathematics

Following up on an earlier post about zero in Greek mathematics and this timeline of zero, I want to say something more about the role of 0 (zero) and 1 (one) in ancient Greek thought. Unfortunately, some of the discussion on Greek mathematics out there is a bit like this:

0 and 1 as quantities

The ancient Greeks could obviously count, and they had bankers, so they understood credits and debts, and the idea of your bank account being empty. However, they had not reached the brilliant insight of Brahmagupta, around 628 AD, that you could multiply a debt (−) and a debt (−) to get a credit (+).

The ancient Greeks had three words for “one” (εἷς = heis, μία = mia, ἑν = hen), depending on gender. So, in the opening line of Plato’s Timaeus, Socrates counts: “One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests … ? (εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων … ; )”

The Greeks had two words for “nothing” or “zero” (μηδέν = mēden and οὐδέν = ouden). So, in the Christian New Testament, in John 21:11, some fisherman count fish and get 153, but in Luke 5:5, Simon Peter says “Master, we toiled all night and took nothing (οὐδὲν)!”

0 and 1 in calculations

In ordinary (non-positional) Greek numerals, the Greeks used α = 1, ι = 10, and ρ = 100. There was no special symbol for zero. Greek mathematicians, such as Archimedes, wrote numbers out in words when stating a theorem.

Greek astronomers, who performed more complex calculations, used the Babylonian base-60 system. Sexagesimal “digits” from 1 to 59 were written in ordinary Greek numerals, with variations of ō for zero. The overbar was necessary to distinguish ō from the letter ο, which denoted the number 70 (since an overbar was a standard way of indicating abbreviations, it is likely that the symbol ō was an abbreviation for οὐδὲν).

Initially (around 100 AD) the overbar was quite fancy, and it became shorter and simpler over time, eventually disappearing altogether. Here it is in a French edition of Ptolemy’s Almagest of c. 150 AD:

In Greek-influenced Latin astronomical calculations, such as those used by Christians to calculate the date of Easter, “NULLA” or “N” was used for zero as a value. Such calculations date from the third century AD. Here (from Gallica) is part of a beautiful late example from around 700 AD (the calendar of St. Willibrord):

Outside of astronomy, zero does not seem to get mentioned much, although Aristotle, in his Physics (Book 4, Part 8) points out, as if it is a well-known fact, that “there is no ratio of zero (nothing) to a number (οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν),” i.e. that you cannot divide by zero. Here Aristotle may have been ahead of Brahmagupta, who thought that 0/0 = 0.

0 and 1 as formal numbers?

We now turn to the formal theory of numbers, in the Elements of Euclid and other works. This is mathematics in a surprisingly modern style, with formal proofs and (more or less) formal definitions. In book VII of the Elements (Definitions 1 & 2), Euclid defines the technical terms μονάς = monas (unit) and ἀριθμὸς = arithmos (number):

1. A monas (unit) is that by virtue of which each of the things that exist is called one (μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται).
2. An arithmos (number) is a multitude composed of units (ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος).

So 1 is the monas (unit), and the technical definition of arithmos excludes 0 and 1, just as today the technical definition of natural number is taken by some mathematicians to exclude 0. However, in informal Greek language, 1 was still a number, and Greek mathematicians were not at all consistent about excluding 1. It remained a number for the purpose of doing arithmetic. Around 100 AD, for example, Nicomachus of Gerasa (in his Introduction to Arithmetic, Book 1, VIII, 9–12) discusses the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512 = α, β, δ, η, ιϛ, λβ, ξδ, ρκη, σνϛ, φιβ) and notes that “it is the property of all these terms when they are added together successively to be equal to the next in the series, lacking a monas (συμβέβηκε δὲ πᾱ́σαις ταῖς ἐκθέσεσι συντεθειμέναις σωρηδὸν ἴσαις εἶναι τῷ μετ’ αὐτὰς παρὰ μονάδα).” In the same work (Book 1, XIX, 9), he provides a multiplication table for the numbers 1 through 10:

The issue here is that Euclid was aware of the fundamental theorem of arithmetic, i.e. that every positive integer can be decomposed into a bag (multiset) of prime factors, in no particular order, e.g. 60 = 2×2×3×5 = 2×2×5×3 = 2×5×2×3 = 5×2×2×3 = 5×2×3×2 = 2×5×3×2 = 2×3×5×2 = 2×3×2×5 = 3×2×2×5 = 3×2×5×2 = 3×5×2×2 = 5×3×2×2.

Euclid proves most of this theorem in propositions 30, 31 and 32 of his Book VII and proposition 14 of his Book IX. The number 0 is obviously excluded from consideration here, and the number 1 is special because it represents the empty bag (even today we recognise that 1 is a special case, because it is not a prime number, and it is not composed of prime factors either – although, as late as a century ago, there were mathematicians who called 1 prime, which causes all kinds of problems):

• If two numbers (arithmoi) by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers (ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει) – i.e. if a prime p divides ab, then it divides a or b or both
• Any composite number is measured by some prime number (ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – i.e. it has a prime factor
• Any number (arithmos) either is prime or is measured by some prime number (ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – this would not be true for 1
• If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it (ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, ὑπ᾽ οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν ἐξ ἀρχῆς μετρούντων) – this is a partial expression of the uniqueness of prime factorisation

The special property of 1, the monas or unit, was sometimes expressed (e.g. by Nicomachus of Gerasa) by saying that it is the “beginning of arithmoi … but not itself an arithmos.” As we have already seen, nobody was consistent about this, and there was, of course, no problem in doing arithmetic with 1. Everybody agreed that 1 + 2 + 3 + 4 = 10. In modern mathematics, we would avoid problems by saying that natural numbers are produced using the successor function S, and distinguish that function from the number S(0) = 1.

The words monas and arithmos occur in other Greek writers, not always in the Euclidean technical sense. For example, in a discussion of causes and properties in the Phaedo (105c), Plato tells us that “if you ask what causes an arithmos to be odd, I shall not say oddness, but the monas (οὐδ᾽ ᾧ ἂν ἀριθμῷ τί ἐγγένηται περιττὸς ἔσται, οὐκ ἐρῶ ᾧ ἂν περιττότης, ἀλλ᾽ ᾧ ἂν μονάς).” Aristotle, in his Metaphysics, spends some time on the philosophical question of what the monas really is.

In general, the ancient Greeks seem to have had quite a sophisticated understanding of 0 and 1, though hampered by poor vocabulary and a lack of good symbols. Outside of applied mathematics and astronomy, they mostly worked with what we would call the multiplicative group of the positive rational numbers. What they were missing was any awareness of negative numbers as mathematical (not just financial) concepts. That had to wait until Brahmagupta, and when it came, 0 suddenly became a whole lot more interesting, because it eventually became possible to define more advanced mathematical concepts like fields.

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Modal logic, knowledge, and an old joke

Recently, I posted about necessary truth and about the logic of belief. I would like to follow up on that by discussing epistemic logic, the logic of knowledge. Knowledge is traditionally understood as justified true belief (more on that below), and we can capture the concept of knowledge using the 4 rules of S4 modal logic. These are in fact the same 4 rules that I used for for necessary truth, and the first 3 rules are the same as those I used for belief (the fourth rule adds the fact that knowledge is true).

Knowledge is specific to some person, and I am replacing the previous modal operators with  Ⓚ  which is intended to be read as “John knows” (hence the K in the circle):

• if P is any tautology, then  Ⓚ P
• if  Ⓚ P  and  Ⓚ (P ⇒ Q)  then  Ⓚ Q
• if  Ⓚ P  then  Ⓚ Ⓚ P
• if  Ⓚ P  then  P

For those who prefer words rather than symbols:

• if P is any tautology, then John knows P
• if John knows both P and (P implies Q), then John knows Q
• if John knows P, then John knows that he knows P
• if John knows P, then P is true

Epistemic logic is useful for reasoning about, among other things, electronic commerce (see this paper of mine from 2000). How does a bank know that an account-holder is authorising a given transaction? Especially if deceptive fraudsters are around? Epistemic logic can highlight which of the bank’s decisions are truly justified. For this application, the first rule (which implies knowing all of mathematics) actually works, because both the bank’s computer and the account-holder’s device can do quite sophisticated arithmetic, and hence know all the mathematical facts relevant to the transaction they are engaged in.

But let’s get back to the idea of knowledge being justified true belief. In his Theaetetus, Plato has Theaetetus suggest exactly that:

“Oh yes, I remember now, Socrates, having heard someone make the distinction, but I had forgotten it. He said that knowledge was true opinion accompanied by reason [ἔφη δὲ τὴν μὲν μετὰ λόγου], but that unreasoning true opinion was outside of the sphere of knowledge; and matters of which there is not a rational explanation are unknowable – yes, that is what he called them – and those of which there is are knowable.” (Theaetetus, 201c)

Although he also uses essentially this same definition in other dialogues, Plato goes on to show that it isn’t entirely clear what kind of “justification” or “reason” is necessary to have true knowledge. In a brief 1963 paper entitled “Is Justified True Belief Knowledge?,” the philosopher Edmund Gettier famously took issue with the whole concept of justified true belief, and provided what seemed to be counterexamples.

My personal opinion, which I have argued elsewhere, is that “justified true belief” works fine as a definition of knowledge, as long as the justification is rigorous enough to exclude beliefs which are “accidentally correct.” For analysing things like electronic commerce, a sufficient level of rigour would involve the use of epistemic logic, as described above.

One of Gettier’s supposed counterexamples involves a proposition of the form  P ∨ Q  (P or Q) such that:

• Smith believes and knows  P ⇒ (P ∨ Q)
• Smith believes P
• P is false
• Q is true, and therefore so is  P ∨ Q

From these propositions we can use doxastic logic to infer that Smith believes the true statement  P ∨ Q,  but we cannot infer (using epistemic logic) that Smith knows  P ∨ Q. A famous old joke is perhaps relevant here:

“A physicist, a philosopher, and a mathematician are travelling through Scotland by train. Through the window, they observe a black sheep in a field. ‘Aha,’ says the physicist, ‘I see that Scottish sheep are black!’ The philosopher responds, ‘No! Some Scottish sheep are black!’ The mathematician, looking shocked, replies: ‘What are you guys saying? All we know is that at least one sheep in Scotland is black on at least one side.’”

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In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation

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Seven varieties of metaphysics

I was having a discussion with someone recently on metaphysics, so I thought I would blog about it. Here are seven varieties of metaphysics, describing three “layers” of reality (and yes, I am oversimplifying for brevity).

The first is Platonism. Plato believed that there was a hierarchy of Forms (Ideals), of which the highest was The One (Plato’s version of God). These Forms or Ideals were the true reality, and the physical objects we touched, saw, and tasted were only shadows of that true reality (that is the point of the allegory of the cave). The physical orange which we see and eat reflects Ideals such as “Fruit,” “Sphere,” and “Orange.” Neoplatonism continues and extends this point of view.

Saint Augustine and many later Christians held to a Christianised Platonism, in which the Ideals were thoughts in the mind of God (the Christian God, naturally). The physical objects we touched, saw, and tasted had a greater importance in Christian Platonism than they did for Plato – after all, when God created those objects, “God saw that it was good.” Much as with Platonism, the regularities that people see in the physical universe are explained by the fact that God created the universe in accordance with regularities in the Divine thoughts. However, Christian Platonism does not have the metaphysical hierarchy that Platonism or Neoplatonism have – in Christian Platonism, God makes direct contact with the physical universe.

Aristotle also reacted to Plato by increasing the importance of the bottom layer, and Aristotle’s thought was Christianised by Thomas Aquinas as Thomism. However, in Thomism the all-important bottom layer does very little except to exist, to have identity, and to have properties assigned to it. It is also not observable in any way. This can be seen in the Catholic doctrine of transubstantiation. According to the Tridentine Catechism of 1566, the bread and the wine of the Eucharist lose their bottom (“substance”) layer (“All the accidents of bread and wine we can see, but they inhere in no substance, and exist independently of any; for the substance of the bread and wine is so changed into the body and blood of our Lord that they altogether cease to be the substance of bread and wine”), while the bottom (“substance”) layer of the body and blood of Christ becomes metaphysically present instead.

Idealism denies that the physical universe exists at all. The followers of Mary Baker Eddy take this view, for example, as did George Berkeley. Only thought exists. To quote a famous movie line, “there is no spoon.” These thoughts may be independent of whatever God people believe in or, as in monistic Hinduism, they may be actually be the thoughts of God (in which case, only God exists).

The last three kinds of metaphysics deny the existence of any kind of God. In Platonist Materialism, this denial is combined with a Platonist approach to mathematics, about which I have written before. Mathematics exists independently of the physical universe, and controls the physical universe, in the sense that the physical universe follows mathematical laws. Roger Penrose is one of many scientists holding this view.

In what I am calling Extreme Materialism, the existence of an independent mathematical world is also denied, i.e. there is an empiricist approach to mathematics (mathematics simply describes observed regularities in nature). This view seems to be increasing in popularity among non-religious people, although it causes philosophical problems for mathematics.

Finally, the concept of the Mathematical Universe holds that the so-called “physical universe” is itself composed only of mathematical objects – only mathematics exists (which makes this, in fact, a kind of Idealism).

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Some Oldest Manuscripts

The chart below (click to zoom) shows the dates of ten significant written works:

• Plato’s Phaedo
• Euclid’s Elements
• Caesar’s Gallic Wars
• The Gospel of John
• Boethius’ Consolation of Philosophy
• Bede’s Historia Ecclesiastica
• Beowulf (date of writing is approximate)
• Dante’s Divine Comedy
• Shakespeare’s Hamlet
• Jane Austen’s Persuasion

Each work is indicated by a vertical line, which runs from the date of writing to the date of the oldest surviving complete copy that I am aware of (marked by a dark circle). Open circles show some of the older partial or fragmentary manuscripts (these act as important checks on the reliability of later copies).

Two threshold periods (marked with arrow) are worth remarking on. First, Gutenberg’s printing press – after its invention, we still have at least one first edition for many important works. Second, the invention of Carolingian minuscule – many older works were re-copied into the new, legible script after that time. They were then widely distributed to monasteries around Europe, so that survival from that period has been fairly good. In the Byzantine Empire, Greek minuscule had a similar effect.

The Bible is a special case (I have highlighted one particular gospel on the chart). It was copied so widely (and so early) that many ancient manuscripts survive.

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MODSIM 2013 Day 5 (Thursday)

MODSIM2013 is continuing in Adelaide. Jerzy Filar (Flinders University) and Russell Glenn (Australian National University) will be giving plenary presentations, and there will be over 200 other talks today.

The first plenary, entitled “The power and limitations of mathematical models and Plato’s Cave Parable,” sounds particularly intriguing, given my interest in Plato and Platonism.

Plato’s Cave (Michiel Coxie, 16^th century)

In the afternoon, I will be chairing a session on Homeland Security and Emergency Management applications of modelling and simulation, which includes six presentations:

• Modeling and Simulation for Homeland Security: K.L. Stamber, T.J. Brown, D.J. Pless, and A. Berscheid
• Algebraic models for path-based measures in time-ordered social networks [full paper elsewhere]: N. Kontoleon, L. Falzon, and P. Pattison
• Using models to compare the effectiveness of alternative strategic security arrangements [abstract only]: R.A. Nunes-Vaz, S. Lord, D. Bilusich, and L. Chim
• An Investigation of the Effectiveness of Interdiction Regimes Against Terrorist Attacks in an Urban Transport Hub: D. Keep, I. Piper, and A. Green
• Microsimulation Study of the Release of Pneumonic Plague and Smallpox on a Synthetic Civilian Population: A. Green, I. Piper, and D. Keep
• Statistics of chemical tracer concentration in a multi-compartment structure measured with a sensor network: S. Karunasekera, A. Skvortsov, A. Gunatilaka, and D. Pitaliadda

The conference dinner will also be held today. That will be fun!

The Torrens River, near the conference venue

Why did Science begin?

Following on from my previous post about the origin of science in the 12^th century, one might ask why the influx of ideas from the Muslim world led to such a scientific explosion in Europe (I’ve been having some Facebook discussions on this). The philosopher Alfred North Whitehead (1861–1947), in his Science and the Modern World (1926, pp 15–16), suggested (perhaps surprisingly) that the credit lay with medieval theology:

“I do not think, however, that I have even yet brought out the greatest contribution of medievalism to the formation of the scientific movement. I mean the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. Without this belief the incredible labours of scientists would be without hope. It is this instinctive conviction, vividly poised before the imagination, which is the motive power of research:—that there is a secret, a secret which can be unveiled. How has this conviction been so vividly implanted on the European mind?

When we compare this tone of thought in Europe with the attitude of other civilisations when left to themselves, there seems but one source for its origin. It must come from the medieval insistence on the rationality of God, conceived as with the personal energy of Jehovah and with the rationality of a Greek philosopher. Every detail was supervised and ordered: the search into nature could only result in the vindication of the faith in rationality. Remember that I am not talking of the explicit beliefs of a few individuals. What I mean is the impress on the European mind arising from the unquestioned faith of centuries. By this I mean the instinctive tone of thought and not a mere creed of words.

In Asia, the conceptions of God were of a being who was either too arbitrary or too impersonal for such ideas to have much effect on instinctive habits of mind. Any definite occurrence might be due to the fiat of an irrational despot, or might issue from some impersonal, inscrutable origin of things. There was not the same confidence as in the intelligible rationality of a personal being. I am not arguing that the European trust in the scrutability of nature was logically justified even by its own theology. My only point is to understand how it arose. My explanation is that the faith in the possibility of science, generated antecedently to the development of modern scientific theory, is an unconscious derivative from medieval theology.”

There are perhaps three relevant theological ideas in the medieval theology to which Whitehead refers. The first is the idea of the Universe as rational, because it is created by a rational God. Such an idea is implicit in, for example, the Timaeus of Plato, which suggests that eternally existing Platonic solids were used by the Creator as the shapes for the different kinds of atom:

The belief in rationality also prompted some good medieval work in the field of logic. However, Whitehead suggests that medieval theology also incorporated “the personal energy of Jehovah.” In particular, the “scrutability of nature” – the idea that Nature is knowable – is implicit in the medieval idea of Nature as a written book, intended to be read. Galileo famously quoted Tertullian (c. 160–225) on this point:

“God is known first through Nature, and then again, more particularly, by doctrine – by Nature in His works, and by doctrine in His revealed Word.” (Tertullian, Against Marcion, I:18; Galileo, Letter to the Grand Duchess Christina of Tuscany, 1615)

Galileo later expanded on the mathematical language in which he thought the “book” of Nature was written:

“La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i carattere, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.

[Science] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” (Galileo, Il Saggiatore, 1623, tr. Stillman Drake)

[This passage has often been quoted, although today we would instead speak of equations and other mathematical constructs.]

Finally, there is the idea that the studying the Universe has value. Whitehead refers to belief systems which considered the world to be unintelligible. There were also other belief systems which devalued even the attempt to understand the world. The Neo-Platonists, for example, focussed their attention on mystical appreciation of the divine things “above,” which left little room for detailed study of the mundane and physical down here “below” (although it did encourage the Neo-Platonists to do mathematical work). The medievals flirted with Christian forms of Neo-Platonism, but the belief that God had created the Universe always gave the mundane and physical its own inherent value, as far as they were concerned.

The Stoics, on the other hand, believed in a cyclic Universe which was periodically destroyed, only for history to repeat itself in exact detail, like a serpent eating its own tail. There is a degree of pointlessness in such a viewpoint which perhaps discourages scientific investigation. Certainly, neither the Neo-Platonists nor the Stoics built the kind of scientific structure that Europeans began to construct in the 12^th century.

Today, of course, the rationality and knowability of the Universe are largely taken for granted (except, perhaps, by Postmodernists), and more people are involved in the scientific enterprise than ever before. The spectacular success of science has made the rationality and knowability of the Universe so obvious, in fact, that it is difficult to comprehend a time, thousands of years ago, when most people thought that unpredictable chaos was all there was.

See also “When Did Modern Science Begin?” by Edward Grant [American Scholar, 66 (1), Winter 1997, 105–113].

Alternatives to mathematical Platonism (2)

In my previous two posts, I outlined the Platonist view of mathematics, and the empiricist alternative. There are also two other alternatives:

Formalism

The truths of mathematics appear to be different in nature from the truths of physics. Formalism accepts this, but suggests that the nature of mathematics is inherently cultural. Different branches of mathematics are essentially just games with symbols and arbitrary rules – games that don’t have any particular meaning. Mathematicians simply work within the chosen rules. However, apart from the problem of the “unreasonable effectiveness of mathematics,” the idea that these rules are chosen arbitrarily runs counter to the experience of most mathematicians. When the concept of “number” was extended to include the imaginary numbers, for example, consistency with the existing rules meant that there was very little choice about how imaginary numbers behaved. In the words of mathematician Jacques Hadamard: “We speak of invention: it would be more correct to speak of discovery… Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray” (from the introduction to The Psychology of Invention in the Mathematical Field).

Many officially formalist mathematicians are Platonists at heart. Jean Dieudonné once wrote with refreshing honesty: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working on something real. This sensation is probably an illusion, but is very convenient.” (from “The Work of Nicholas Bourbaki,” American Mathematical Monthly, 77(2), Feb 1970, p. 134–145).

Logicism

Most mathematicians feel that the truths of mathematics are indeed in a different category from the truths of physics – that the truths of mathematics in a sense come first. Logicism is a way of rescuing this aspect of Platonism while avoiding the more mystical aspects. The basis for logicism is that logic also comes before physics – all sciences assume logical thought as a starting point. Logical truths “exist” in some sense, and logicists assume that there are no philosophical difficulties about this kind of existence. In other words, a mystical Platonic world is not needed to explain logic. Consequently, if we can provide a foundation for mathematics in terms of pure logic, we can retain all the benefits of Platonism without any of the problems.

In logicism, numbers are defined as being particular kinds of sets. Logicism began with Gottlob Frege, who published two volumes of his Die Grundgesetze der Arithmetik in 1893 and 1903. Sadly for Frege, his fellow mathematician and philosopher Bertrand Russell found a major flaw – now known as “Russell’s paradox” – in the work, just before the second volume was published. With Alfred North Whitehead, Russell was able to repair the flaw, in a three-volume work called Principia Mathematica (published in 1910, 1912, and 1913).

After 379 pages, Whitehead and Russell are well on the way to proving that 1 + 1 = 2.

The logicist programme, however, is not free of problems. First, it is extremely complex. It took Whitehead and Russell hundreds of pages of complicated logic to prove that 1 + 1 = 2. Normally, we try to explain complex things in terms of simple ones. It seems a little perverse to give such a complicated explanation of numerical facts that we understood in kindergarten. And, by including set theory as part of the basis, it isn’t really “pure logic” any more.

Second, there is more than one way of defining numbers as sets, and none of them is obviously “right.” This has led to the suggestion that sets are “what numbers could not be” (the title of an article by Paul Benacerraf in The Philosophical Review, 74, Jan 1965, pp. 47–73), and that numbers must be fundamentally different in nature from sets – if not Platonic objects satisfying certain axioms, then something else which exists in a non-contingent way.

Third, it is unclear whether logicism has actually gained anything. The starting assumption was that logic was simple and obvious, raising no philosophical problems. But if all of mathematics is hidden deep inside the structure of logic, then perhaps logic is not as simple as it first seemed. Mathematics and logic may in fact be different aspects of the same thing, but this may not make the fundamental questions about mathematical existence go away.

Personally, I still see Platonism as the best answer. How about you?

Alternatives to mathematical Platonism (1)

In my last post, I outlined the view of mathematical Platonism taken by Roger Penrose and other mathematicians. Briefly, in the words of Joel Spencer, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).

Uncovering hidden jewels (James Tissot).

However, some modern mathematicians feel that the time for such ideas is past. In the words of Brian Davies, “It is about time that we … ditched the last remnant of this ancient religion” (in his article “Let Platonism die,” European Mathematical Society Newsletter, June 2007).

One alternative commonly presented is Empiricism. Santa Claus and the Tooth Fairy do not exist, and neither do infinite decimals, or perfect circles, or the set of all natural numbers. Only the physical universe exists. In the words of astronomer Carl Sagan, “The Cosmos is all that is or ever was or ever will be” (the opening sentence of his book Cosmos). Our only truly certain knowledge is physics, and mathematics is in fact a branch of physics. When we say that 2 + 2 = 4, we are not talking about a relationship between Platonic number-objects. Instead, what we mean – and all that we mean – is the empirical truth that two atoms plus two atoms gives four atoms, and likewise for stars, rocks, or people. “Four” is not a noun, it’s an adjective.

The empiricist point of view seems to solve the mystery of the “unreasonable effectiveness of mathematics.” Mathematics is just part of physics, and it isn’t surprising that different branches of physics agree with each other. Empiricism also avoids the need to postulate a “soul” or some other mechanism for peering into an ethereal Platonic world.

Conic sections in theory and practice.

There are two problems with the empiricist point of view, however. First, it isn’t true to the history of mathematics. Galileo used parabolas to describe the motion of falling objects, but the ancient Greeks had originally described parabolas in a quite different context, that of conic sections. Similarly, imaginary numbers were originally discussed without the slightest idea that centuries later they would become a fundamental part of quantum theory. There’s still a mystery there: why should mathematics from one context work so well in another?

The second problem is that scientific truths are contingent. We could, for example, live in a world where plants were purple. The fundamental forces of physics could be different from the way they are (although that could imply a lifeless universe). We can even imagine a universe containing no matter at all, only empty space. The laws of mathematics, however, could not have been different – they are necessary. It is impossible to imagine – at least, to imagine consistently – a universe in which 2 + 2 = 5. Even God cannot make 2 + 2 = 5. This indicates that there must be more to mathematical truth than just physics. Statements about mathematics, such as “2 + 2 = 4” are in a different category to statements about the universe, such as “light travels at 299,792,458 metres per second.” But if that is the case, then mathematical truth must in some sense lie outside the universe – which brings us back to Platonism.

Three Worlds

Roger Penrose, in his book Shadows of the Mind, outlines an idea adapted from Karl Popper – that there are “three worlds.” The physical universe needs no explanation, except perhaps to Bishop Berkeley, while the subjective world of our own conscious perceptions is one we each know well. The third world is the Platonic world of mathematical objects.

Penrose says of the third world: “What right do we have to say that the Platonic world is actually a ‘world,’ that can ‘exist’ in the same kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag-bag of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests on the profound, timeless, and universal nature of these concepts, and on the fact that their laws are independent of those who discover them. The rag-bag – if indeed that is what it is – was not of our creation. The natural numbers were there before there were human beings, or indeed any other creature here on earth, and they will remain after all life has perished.” (Shadows of the Mind, p. 413)

Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it more poetically: “In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.” G. H. Hardy was ambivalent about the Divine, but like most mathematicians he believed “that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.”

This trilogy of worlds raises some questions, of course. The first is what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” As William Newton-Smith asks, “if mathematics is about this independently existing reality, how come it is useful for dealing with the world?” Why does the world follow the dictates of eternal Reason? Or, as Einstein put it, “how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

The second question is the mind-body problem. How are we conscious of the universe, and how do our decisions to act affect it? Does even our perception have strange quantum effects?

Finally, how do we become aware of the Platonic world? Elsewhere, Penrose says “When one ‘sees’ a mathematical truth, one’s consciousness breaks through into this world of ideas, and makes direct contact with it… When mathematicians communicate, this is made possible by each one having a direct route to truth” (The Emperor’s New Mind, p. 554). But what exactly does that mean? Does one’s soul go on some kind of “spirit journey”?

Doubling the square

Plato, in the story of Socrates and Meno’s slave, tells how an uneducated slave is prompted to discover how to double a square. Plato saw this as evidence of memory from a past life, but it provides an example of mathematical intuition that all (successful) students of mathematics will recognise. As Saint Augustine said, “The man who knows them [mathematical lines] does so without any cogitation of physical objects whatever, but intuits them within himself.” Yet Plato’s (and Augustine’s) belief in such an intuitive soul makes the mind-body problem more acute. How do the three worlds tie together? It seems a mystery.

Three Worlds by M. C. Escher

Why study mathematics?

Like John Allen Paulos, I am often asked why mathematics is worth studying. In his book A Mathematician Reads the Newspaper (Basic Books, 1995), Paulos gives an excellent answer:

“As a mathematician, I’m often challenged to come up with compelling reasons to study mathematics. If the questioner is serious, I reply that there are three reasons or, more accurately, three broad classes of reasons to study mathematics. Only the first and most basic class is practical. It pertains to job skills and the needs of science and technology. The second concerns the understandings that are essential to an informed and effective citizenry. The last class of reasons involves considerations of curiosity, beauty, playfulness, perhaps even transcendence and wisdom.”

The second and third answers are reflected in the words inscribed on the door of Plato’s Academy: “Let no one ignorant of geometry enter” (Ἀγεωμέτρητος μηδεὶς εἰσίτω):

The first answer relates to the critical importance of mathematics in several fields of human endeavour, including science, engineering, medicine, and finance. For example:

A stressed ribbon bridge is strong if its shape is that of the mathematical curve called a catenary.

The spread of an infectious disease can be predicted by a set of three differential equations, relating three variables: S, I, and R (left). Real-world disease outbreaks show a similar pattern (right).

Many people list this as the only reason for studying mathematics, but it only applies to a minority of students – those keeping open the option of entering those fields. The second answer relates to the importance of mathematics in decision-making by ordinary citizens, and this applies to everybody. Some of those decisions by citizens require quantitative thinking. For example, which groceries are the best value for money? If two studies on 20 people report that a certain vegetable causes cancer, and one study on 1,000 people report that it doesn’t, is the vegetable safe? More subtly, training in mathematics helps in thinking clearly even about non-quantitative issues. Plato seemed to think that mathematics was essential training, and I would agree. Bertrand Russell put it this way: “One of the chief ends served by mathematics, when rightly taught, is to awaken the learner’s belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration.”

What is the best value for money – the melons at $5 each, the grapes at $4 per kg, or the blueberries at $3 per punnet?

This classic book has applications to more than mathematics.

The third answer relates to a famous remark of Debussy – “La musique est une mathématique mystérieuse dont les éléments participent de l’infini” (“Music is a mysterious mathematics whose elements partake of the Infinite”). It works the other way around too. Mathematics is a mysterious and beautiful music that puts one in touch with the Infinite. As Plato would have said, mathematics reminds us that more things exist than just the finite and physical. This particularly applies to those parts of mathematics which relate to infinity, such as the number π, or the Mandelbrot set:

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 …

Some of the (infinitely many) digits of π.

The Mandelbrot set contains an infinite amount of detail (click for zoom animation).

Rudy Rucker’s little book The Fourth Dimension and How to Get There is also a great mind-stretcher. And, of course, having one’s mind stretched like that is a lot of fun.