]]>

]]>

]]>

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)!”

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.

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):

- 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). - 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.

]]>

Following up on an earlier post about Zero in Greek mathematics, here is a timeline for the use of zero in Europe (click to zoom). I have used images of, or quotes from, primary sources where possible (reliably dated Indian primary sources are much harder to find than Greek ones, unfortunately).

Chinese uses of zero are probably also derived from the Greeks, but Mayan uses are clearly independent.

]]>

The chart above shows national Covid mortality against latitude of national capitals (open circles are for the Southern Hemisphere, solid circles for the Northern). The trend line in blue has a correlation of 0.50 (with *p* < 10^{−13}). Countries further away from the equator are definitely reporting more Covid deaths.

It is possible that these numbers reflect under-counting in the tropics (although this is unlikely for Singapore = SG) and over-counting in wealthier countries away from the tropics (e.g. by reporting deaths of patients with positive Covid tests as Covid deaths, even if the actual cause of death is unrelated). However, it seems unlikely that under-counting and over-counting can explain everything here.

This paper in *The Lancet* notes that “It has long been clear that groups that traditionally exhibit vitamin D deficiency or insufficiency, such as older adults and nursing home residents, and Black, Asian, and minority ethnic populations, are the same groups that have also been disproportionately impacted by COVID-19. Additionally, increased time spent indoors due to strict lockdowns and shielding triggered concerns that some people might not obtain the necessary physiological levels of vitamin D from sunlight.”

My chart above is consistent with this: decreased sunshine away from the equator appears to increase Covid mortality, presumably due to vitamin D deficiency. This study in *QJM* notes, “vitamin D supplementation is effective in reducing COVID-19 severity. Hence vitamin D should be recommended as an adjuvant therapy for COVID-19.” Personally, I have been taking this advice for quite some time.

]]>

- the fetal or embryonic heart begins beating at about 21 days after conception, as any textbook will point out
- as the table shows, after a few weeks, the fetus or embryo is
**not**“too small to see with the naked eye”

Fetal length data in the table is mostly from here. Except where indicated, linked images are subsequent to miscarriage or to surgery to resolve ectopic pregnancy, so may be distressing to some readers.

Week post fertilisation | Week post LMP | Fetal length | Image |
---|---|---|---|

1 | 3 | 0.01 cm / 0.005 inches | 8–cell image |

2 | 4 | 0.02 cm / 0.008 inches | – |

3 | 5 | 0.1 cm / 0.04 inches | heart begins beating at 21 days |

4 | 6 | 0.5 cm / 0.2 inches | image on flickr |

5 | 7 | 1 cm / 0.4 inches | image on wikimedia |

6 | 8 | 1.6 cm / 0.6 inches | image on flickr |

7 | 9 | 2.3 cm / 0.9 inches | image on flickr |

8 | 10 | 3.2 cm / 1.3 inches | – |

9 | 11 | 4.1 cm / 1.6 inches | – |

10 | 12 | 5.4 cm / 2.1 inches | – |

11 | 13 | 6.7 cm / 2.6 inches | ultrasound image |

12 | 14 | 14.7 cm / 5.8 inches | – |

13 | 15 | 16.7 cm / 6.6 inches | – |

14 | 16 | 18.6 cm / 7.3 inches | – |

15 | 17 | 20.4 cm / 8 inches | ultrasound image |

16 | 18 | 22 cm / 8.7 inches | – |

Below (from here) is a chart of heart development:

]]>

- if
*P*is any tautology, then Ⓞ*P* - if Ⓞ
*P*and Ⓞ (*P*⇒*Q*) then Ⓞ*Q* - if Ⓞ
*P*then Ⓞ Ⓞ*P* - if Ⓞ
*P*then ~ Ⓞ ~*P*

where ~ Ⓞ ~*P* is read as “~ *P* is not obligatory,” i.e. “*P* is permissible.” For those who prefer words rather than symbols:

- if
*P*is any tautology, then*P*is obligatory - if
*P*and (*P*implies*Q*) are both obligatory, then*Q*is obligatory - if
*P*is obligatory, then it is obligatory that*P*is obligatory - if
*P*is obligatory, then*P*is permissible

For these rules as they stand, the only things that are obligatory are necessary truths like 2 + 2 = 4. This is because **you can’t get an “ought” from an “is.”** Apart from the first rule, there is no way of introducing a Ⓞ symbol out of nowhere. Consequently, if we are to reason about ethics and morality, we must begin with some **deontic axioms** that already contain the Ⓞ symbol. For people of faith, these deontic axioms may be given by God, as in the 10 Comandments, which include:

Ⓞ you do not murder.

Ⓞ you do not commit adultery.

Ⓞ you do not steal.

Ⓞ you do not bear false witness against your neighbor.

Immanuel Kant famously introduced the categorical imperative, a deontic axiom which Kant thought implied all the other moral rules, and thus provided the smallest possible set of deontic axioms:

Ⓞ [you] act only according to that maxim whereby you can, at the same time, will that it should become a universal law.

Others have suggested the greatest happiness of the greatest number as a principle. Fyodor Dostoevsky, William James, and Ursula Le Guin are among those who have explained the problem with this:

“Tell me yourself, I challenge your answer. Imagine that you are creating a fabric of human destiny with the object of making men happy in the end, giving them peace and rest at last, but that it was essential and inevitable to torture to death only one tiny creature – that baby beating its breast with its fist, for instance – and to found that edifice on its unavenged tears, would you consent to be the architect on those conditions?” (Fyodor Dostoevsky, “The Grand Inquisitor,” in

The Brothers Karamazov, 1880; 4.35 on Goodreads)

“Or if the hypothesis were offered us of a world in which Messrs. Fourier’s and Bellamy’s and Morris’s Utopias should all be outdone and millions kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture, what except a specifical and independent sort of emotion can it be which would make us immediately feel, even though an impulse arose within us to clutch at the happiness so offered, how hideous a thing would be its enjoyment when deliberately accepted as the fruit of such a bargain?” (William James, “The Moral Philosopher and the Moral Life,” 1891)

“Some of them understand why, and some do not, but they all understand that their happiness, the beauty of their city, the tenderness of their friendships, the health of their children, the wisdom of their scholars, the skill of their makers, even the abundance of their harvest and the kindly weathers of their skies, depend wholly on this child’s abominable misery.” (Ursula K. Le Guin, “The Ones Who Walk Away from Omelas,” 1973; reprinted in

The Wind’s Twelve Quarters, 1975; 4.05 on Goodreads)

The **meaning** of deontic statements can be described using Kripke semantics, which exploits the idea of **possible worlds** (i.e. alternate universes). To say that some statement is obligatory is to say that the statement would be true in **better possible worlds** (we write *w*_{1} → *w*_{2} to mean that *w*_{2} is a better possible world than *w*_{1}).

In any given world *v*, the statement Ⓞ *P* is equivalent to :

*P*is true in all better worlds*w*(i.e. all those with_{i}*v*→*w*)_{i}

Likewise, in any given world *v*, the statement ~ Ⓞ ~*P* (*P* is permissible) is equivalent to:

*P*is true in at least one better world*w*(i.e. one with_{i}*v*→*w*)_{i}

The rules of deontic logic imply two conditions on these arrows between possible worlds:

- if
*w*_{1}→*w*_{2}→*w*_{3}then*w*_{1}→*w*_{3}(i.e. chains of arrows are treated like arrows too) - in every world
*v*there is at least one arrow*v*→*w*(i.e. chains of arrows don’t stop; this includes the case of*v*→*v*)

A number of philosophers have suggested that deontic logic leads to paradoxes. In all cases that I have seen, these “paradoxes” have involved simple errors in the use of deontic logic – errors that become obvious when the deontic statements are translated into statements about possible worlds.

There are limitations to deontic logic, however. For example, if we say that it is obligatory not to steal, this means that, in all better possible worlds, nobody steals. If we also say that it is obligatory to punish thieves, this means that, in all better possible worlds, thieves are punished. However, if it is obligatory not to steal, better possible worlds have no thieves, so the two statements do not combine well.

Some people would, no doubt, suggest that fiction like that of Dostoevsky is a better tool than logic for exploring such issues. In cases where the writer is a genius, they are probably right.

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

]]>

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!

SomeScottish sheep are black!’ The mathematician, looking shocked, replies: ‘What are you guyssaying? All we know is that at least one sheep in Scotland is black on at least one side.’”

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

]]>

Since beliefs are specific to some believing person, I am replacing the modal operator □ with Ⓙ which is intended to be read as “John believes” (hence the J in the circle):

- if
*P*is any tautology, then Ⓙ*P* - if Ⓙ
*P*and Ⓙ (*P*⇒*Q*) then Ⓙ*Q* - if Ⓙ
*P*then Ⓙ Ⓙ*P*

For those who prefer words rather than symbols:

- if
*P*is any tautology, then John believes*P* - if John believes both
*P*and (*P*implies*Q*), then John believes*Q* - if John believes
*P*, then John believes that he believes*P*

These rules are very useful for helping computer systems (such as autonomous vehicles) reason about the beliefs of other entities (“If John believed a car was coming, he would not cross the road. But he is crossing the road. Therefore he does not believe that a car is coming. We should warn him.”).

As stated above, however, the rules are extremely optimistic about John’s knowledge of mathematics and logic. For some applications, we may need to assume that John believes less of that stuff. There is also a problem in assuming that John accepts the logical consequences of his beliefs. Real people do not always do that. Some years ago, I posted about the idea of a flat earth (an idea that medieval people were too wise to accept). Most believers in a “flat earth” do **not** accept the logical consequences of their beliefs. In particular, for the most popular “flat earth” model, sunsets would never be observed, because the sun always remains above the “disc of the earth.” Flat-earthers refuse to admit such consequences. Common air travel routes to and from Australia would also be impossible (see below), but flat-earthers generally realise the incompatibility of those routes, and simply deny that they exist. Logic is perhaps not the best tool for describing such patterns of thought.

The **meaning** of doxastic statements can be described using Kripke semantics, which exploits the idea of **possible worlds** (i.e. alternate universes). To say that John believes some statement is to say that the statement is true in the alternate universes that John thinks he might be living in. In those alternate universes, the earth might indeed be flat.

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

]]>