9 Numbers

9 Numbers
Note:9@@@@@@@@

Yellow highlight | Location: 2,500

9.1   Cardinal numbers as properties
Note:Ttttttt

Yellow highlight | Location: 2,501

what is a number?
Note:IN CERCA DI TEORIE ALTERNATIVE

Yellow highlight | Location: 2,502

A cardinal number is a kind of property.1 These properties are best ‘defined ostensively’, that is, by giving examples.
Note:DEF...SI PARTE DA ESEMPI

Yellow highlight | Location: 2,504

I would show them something like Figure 9.1. In that picture, there are two stars, two hexagons, and two lightning bolts.
Note:COME INSEGNARE IL NUMERO 2

Yellow highlight | Location: 2,507

It is no coincidence that this is how children are actually taught
Note:BAMBINI

Yellow highlight | Location: 2,508

The ontological status of the number two is thus the same as that of other universals,
Note:ONTOLOGIA

Yellow highlight | Location: 2,512

9.2   Frege’s objection
Note:Ttttttttt

Yellow highlight | Location: 2,514

the same concrete thing can be said to instantiate different numbers. Suppose you have a deck of cards. What is the number that it instantiates? Well, it is one deck, but it is fifty-two cards.
Note:OBIEZIONE DEL MAZZO DI CARTE

Yellow highlight | Location: 2,517

Therefore, number is not a property of a concrete object, such as the deck of cards. Instead, Frege proposes, numbers must be properties of ‘concepts’

Note | Location: 2,517

SOLUZIONE FREGE...OB IDWALISTA...ACCETFABILW IMO

Yellow highlight | Location: 2,519

we resolve the puzzle of the deck of cards by saying that there are two distinct concepts: the concept ‘deck’ (or maybe, ‘deck that is on this table now’) and the concept ‘card’ (or, ‘card that is on this table now’),
Note:SOLUZIONE

Yellow highlight | Location: 2,521

Cantor and Russell, on the other hand, would ditch the talk of concepts and say that there are two distinct sets:
Note:SOLUZIONE RUSSELL

Yellow highlight | Location: 2,525

Each number is a property of a concrete particular
Note:L ASSUNTO CONSIDERATO DA FREGE

Yellow highlight | Location: 2,530

there is only one number that is capable of being a property of a concrete particular, and that is the number 1. It is logically impossible, for example, for the number 2 to be a property of a concrete object; the number 2 can only be a property of two concrete objects.
Note:L ASSUNTO CORRETTO E IMPERMEABILE ALLA CRITICA DI FREGE

Yellow highlight | Location: 2,537

it is not that twoness applies to the set {a,b};
Note:CccccccccFACCIAMO A MENO DEGLI INSIEMI

Yellow highlight | Location: 2,549

If we’re talking about the aggregate of the cards, that instantiates the number 1.
Note:QUINDI

Yellow highlight | Location: 2,552

9.3   Arithmetical operations
Note:Tttttttttttt

Yellow highlight | Location: 2,553

‘Two apples plus three apples make five apples’ means something like this: if you have two apples, and you also have three more apples (that is, three that are each different from either of the original two), then you have five apples.
Note:ADDIZIONE....E COSÌ LE ALTRE

Yellow highlight | Location: 2,563

It is not, for example, a matter of bringing the apples into spatial proximity.
Note:L OPERAZIONE NN È FISICA

Yellow highlight | Location: 2,566

Adding two apples to three apples is solely a matter of considering two apples and, without making any changes to any of the apples, considering an additional three apples,
Note:OPERAZIONE MENTALE

Yellow highlight | Location: 2,567

There is not even any passage of time assumed:
Note:TEMPO

Yellow highlight | Location: 2,570

This is why arithmetical statements are necessary and knowable a priori.
Note:A PRIORI

Yellow highlight | Location: 2,571

9.4   The laws of arithmetic
Note:Tttttttttt

Yellow highlight | Location: 2,573

a+b = b+a    (Commutative Law of Addition)
Note:LA PIÙ FAMOSA DELLA LEGGI

Yellow highlight | Location: 2,579

Why do these hold, and how do we know them? In essence, the reason why (a + b) is equal to (b + a) is that the expressions ‘(a + b)’ and ‘(b + a)’ are synonymous:
Note:SINONIMIA

Yellow highlight | Location: 2,608

9.5   Zero
Note:Tttttttttt

Yellow highlight | Location: 2,610

If the number n is instantiated by n things, then the number zero must be instantiated by zero things. But there cannot be a property that,
Note:OBIEZIONE

Yellow highlight | Location: 2,612

Reply: zero is not the same kind of thing as one, two, three, and so on.
Note:RISPOSTA

Yellow highlight | Location: 2,613

zero is not a number.
Note:INFATTI

Yellow highlight | Location: 2,616

if I have zero cookies, I should not say, ‘I have a number of cookies’;
Note:SENSO COMUNE

Yellow highlight | Location: 2,617

It is no coincidence that the concept of zero has a quite different history
Note:INFATTI

Yellow highlight | Location: 2,633

Objection: ‘But we can do arithmetical operations using zero. How could that be, if zero is not a true number,
Note:OBIEZIONE

Yellow highlight | Location: 2,635

When we decide to extend the number system by including zero, we define arithmetical operations for zero in such a way as to keep the whole system coherent.
Note:RISPOSTA...SEMPLIFICAZIONE

Yellow highlight | Location: 2,641

Famously, there is one case in which we do not define the arithmetical operations for zero, namely, the case of division.
Note:E LE ECCEZIONI NN MANCANO...A RIBADIRE CHE...

Yellow highlight | Location: 2,648

what is the referent of ‘0’ in its noun usage?

Note | Location: 2,649

LA DOMANDA CHE RESTA

Yellow highlight | Location: 2,649

The answer is that the symbol does not require a referent to be meaningful. In the same way that the noun ‘nothing’ lacks a referent,
Note:RISPISTA

Yellow highlight | Location: 2,653

The point is simply that ‘zero’ functions differently in some important ways from ‘one’, ‘two’,
Note:NN SI TRATTA DI VOLER ESPELLERE LO ZERI DAI NUMER

Yellow highlight | Location: 2,655

9.6   A digression on large numbers
Note:Ttttttt

Yellow highlight | Location: 2,663

there are about 1080 atoms in the observable universe;
Note:STENOGRAFIA

Yellow highlight | Location: 2,665

Now we can define another operation known as ‘tetration’, symbolized by ‘↑↑’, to represent repeated exponentiation.
Note:X È UN RIPETUTO + E ESP È UN RIPETUTO X

Yellow highlight | Location: 2,671

After that, there is the operation of pentation (symbolized by ‘↑↑↑’), which is repeated tetration.
Note:OLTRE

Yellow highlight | Location: 2,707

9.7   Magnitudes and real numbers
Note:Ttttttttttt

Yellow highlight | Location: 2,708

9.7.1   Magnitudes vs. numbers
Note:Tttttttt

Yellow highlight | Location: 2,709

The concept of magnitude is probably undefinable,
Note:OCCORRONO ESEMPI

Yellow highlight | Location: 2,710

the height of the Eiffel Tower, the temperature of a cup of coffee, the mass of the Milky Way Galaxy.
Note:ESEMPI

Yellow highlight | Location: 2,711

properties of concrete objects.
Note:MAGNITUDO

Yellow highlight | Location: 2,713

refers to a dimension
Note:Ccccc

Yellow highlight | Location: 2,714

refers to a particular value
Note:Ccccccc

Yellow highlight | Location: 2,717

Magnitude values are represented using real numbers. The magnitudes, however, are not themselves numbers.
Note:M E N

Yellow highlight | Location: 2,718

The Eiffel Tower is approximately 324 meters tall, which is to say about 1063 feet tall. If the height of the Tower is a number, which one is it? Is it around 324, or around 1063?
Note:QUI FREGE HA RAGIONE

Yellow highlight | Location: 2,722

the number that we use to represent the height of the Tower is not a property of the tower. But the height is a property of the tower. So the height is not a number.
Note:ALTEZZA E NUMERO

Yellow highlight | Location: 2,724

9.7.2   Real numbers as relationships
Note:Tttttttt

Yellow highlight | Location: 2,724

What, then, is a real number?
Note:PRIMA AVEVAMO DEFINITO I N NATURALI COME UNA PROP DEGLI OGGETTO...ORA DEFINIREMO I REALI COME UNA RELAZIONE...I NUMERI QUANDO ESPRIMONO UN UNITÀ DI MISURA

Yellow highlight | Location: 2,727

start with how we use real numbers to describe the world.
Note:PROCEDIMENTO

Yellow highlight | Location: 2,727

Eiffel Tower is 324 meters tall.
Note:2

Yellow highlight | Location: 2,728

‘324’ records a relationship between the height of the Tower and a unit,
Note:ALTEZZA E METRI

Yellow highlight | Location: 2,732

The metaphysical background is that magnitude values come in classes that are comparable to each other.
Note:CONFRONTABILITÀ XFETTA

Yellow highlight | Location: 2,736

When a magnitude is greater than another magnitude, this greaterness comes in degrees.
Note:NUMERI E MAGNITUDO

Yellow highlight | Location: 2,744

9.7.3   In defense of the bifurcated account of number
Note:Tttttttttttt

Yellow highlight | Location: 2,748

‘324 is a real number, since an object could be exactly 324 times as long as the standard meter. 324 is also a natural number. But on your account, a natural number is a property, whereas a real number is a relationship.
Note:OBIEZIONE DELLA BIFORCAZIONE

Yellow highlight | Location: 2,751

It turns out that there are two numbers 324: the cardinal number and the real number.
Note:RISPOSTA

Yellow highlight | Location: 2,756

Now you might wonder: why not take a uniform view of cardinal numbers and real numbers?
Note:DUBBIO

Yellow highlight | Location: 2,758

Frege’s deck-of-cards problem by saying that the same physical aggregate is differently related to two different units: it bears the ‘52-fold’ relationship to the unit ‘card’, while bearing the ‘one-fold’ relationship to the unit ‘deck’.
Note:SOLUZIONE SEMPLICE

Yellow highlight | Location: 2,762

The relationship theory requires that there be an object to be the first relatum, that is, the thing that is supposed to stand in a relationship to a unit.
Note:INCONVENIENTE

Yellow highlight | Location: 2,768

suppose there are seven reasons for being suspicious of set theory.
Note:DOV È QUI L OGGETTO?

Yellow highlight | Location: 2,769

an object that has each of the seven reasons as parts?
Note:L OGGETTO MISTERIOSO....SETTE RAGIONI SETTE METRI

Yellow highlight | Location: 2,783

9.7.4   Measures and magnitudes
Note:Tttttttttttttrr

Yellow highlight | Location: 2,784

measure theory,
Note:BRANCA DELLA MATEMATICA

Yellow highlight | Location: 2,818

9.7.5   Intensive vs. extensive magnitudes
Note:Tttttttt

Yellow highlight | Location: 2,820

Extensive magnitudes are those that are additive across the parts of an object.
Note:ESTENSIVE

Yellow highlight | Location: 2,822

the magnitude values of all of these parts will contribute additively to the magnitude value of the whole.
Note:Ccccccccccc

Yellow highlight | Location: 2,823

length is an extensive magnitude.
Note:ESEMPIO

Yellow highlight | Location: 2,824

Similarly for volume:
Note:ALTRO ESEMPIO

Yellow highlight | Location: 2,827

Intensive magnitudes, by contrast, are magnitudes that do not arise from adding together the magnitudes of the parts of an object,
Note:INTENSIVE

Yellow highlight | Location: 2,828

For example, the temperature
Note:ESEMPIO

Yellow highlight | Location: 2,831

There may be ambiguous cases. Consider the property of mass.
Note:A METÀ

Yellow highlight | Location: 2,839

9.7.6   Natural vs. artificial magnitudes
Note:Ttttttttt

Yellow highlight | Location: 2,841

schmass of an object is defined to be the reciprocal of 3 minus the object’s mass in kilograms:
Note:MAGNITUDO INVENTATA...SI PUÒ

Yellow highlight | Location: 2,846

I will soften that a bit and simply call schmass an ‘artificial’ magnitude,
Note:ARTIFICIALE

Yellow highlight | Location: 2,848

As a first approximation, I will take causal efficacy as the test of naturalness.
Note:SEMPLICITÀ

Yellow highlight | Location: 2,863

9.8   Indexing uses of numbers
Note:Ttttttttttt

Yellow highlight | Location: 2,865

suppose you are in a hotel, in room 210.
Note:NUMERO INDICE

Yellow highlight | Location: 2,866

indicating that there are 210 of something;
Note:NUMERO NATURALE?

Yellow highlight | Location: 2,866

indicating that something is 210 times greater than
Note:NUMERO REALE?

Yellow highlight | Location: 2,868

‘210’ is simply being used as a name for that room.
Note:USO IN QS CASO

Yellow highlight | Location: 2,870

a number term being used in a way that does not really refer to a number.
Note:CONCLUSIONE

Yellow highlight | Location: 2,892

This is what I refer to as an indexing use of numbers. In an indexing use, some number is used as a name for a more or less arbitrary item or value,
Note:INDEXING

Yellow highlight | Location: 2,903

9.9.1   Rational vs. irrational numbers
Note:Tttttttt

Yellow highlight | Location: 2,911

9.9.2   Negative numbers
Note:Ttttttt

Yellow highlight | Location: 2,924

9.9.3   Imaginary numbers
Note:Ttttttttt

Yellow highlight | Location: 2,926

x2 = −1’.
Note:NUMERI IMMAGINARI

Yellow highlight | Location: 2,929

One cannot conjure such a number into existence just by fiat, as the standard approach seems to suppose. Trying to solve the equation ‘x2 = −1’ by inventing a new number is like trying to solve the equation ‘x + 1 = x’ by inventing a new number,
Note:DIFFICILI DA CONCEPIRE

Yellow highlight | Location: 2,943

9.9.4   Infinitesimal numbers
Note:Tttttt

Yellow highlight | Location: 2,944

An infinitesimal number is a number that is smaller than any positive real number but yet greater than zero.
DEF @@@@@@@@@