7 Philosophical Preliminaries

7 Philosophical Preliminaries
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7.1   Metapreliminaries 7.1.1   Why a chapter of preliminaries?
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This chapter explains some general features of my philosophical outlook,
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Readers sympathetic to empiricism will tend to reject the rest of this book.
Note:PREVISIONE

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7.1.2   Modern empiricism
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it is impossible to attain any substantive knowledge of the world except on the basis of observation.
Note:DOGMA EMPIRICO

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rationalism holds that there is some substantive knowledge about the world that is not based on observation;
Note:CONTROPARTE

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pure reason.
Note:ESISTE SOLO X I RAZIONALISTI

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an extreme form of empiricism, logical positivism,
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literally meaningless,
Note:TUTTE LE VERITÀ NN OSSERVABILI

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wide swathes of human intellectual endeavor meaningless, including such fields as metaphysics, theology, and ethics.
Note:CONCLUSIONE RADICALE

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if any scientific theory entails the existence of observationally undetectable facts, that theory must be either false or meaningless.
Note:COROLLARIO IMBARAZZANTE

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empiricists have generally treated their philosophy more like an a priori axiom than like an empirical hypothesis.
Note:IRONIA

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assumed as self-evident,
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Peter van Inwagen describes an ideology of ‘scientism’, consisting in ‘an exaggerated respect for science
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all knowledge must be like scientific knowledge,
Note:L OBBIETTIVO DA XSEGUIRE

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all books not containing mathematics or science should be burned,
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7.1.3   The significance of empiricism
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Positivistic assumptions were crucial in motivating Einstein’s Theory of Relativity, the Copenhagen Interpretation of quantum mechanics, and the formalist approach to mathematics,
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Today, even those who would explicitly disavow positivism tend to accept positivist-inspired theories without knowing their provenance.
Note:TUTTI VITTIME

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some very broad philosophical viewpoints, such as ‘naturalism’.
Note:INFLUENZA INDIRETTA MA XSISTENTE

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Empiricism and positivism set the agenda
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the leading philosophical tradition in the English-speaking world.
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7.2   Phenomenal conservatism
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Phenomenal conservatism holds that undefeated appearances are a source of justification (perhaps the only source of justification) for belief.
Note:GNOSEOLOGIA DEL SENSO COMUNE...PRINCIPIO DI CREDULITÀ

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An appearance is a broad type of mental state, distinct from and normally prior
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including sensory experiences, memory experiences, and rational intuitions.
Note:I VARI TIPI

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When I look out the window, it seems to me that there is a squirrel in the tree (visual appearance); I seem to recall falling off my motorcycle a few times in college (memory appearance); it seems to me that the shortest path between any two points must be a straight line (rational intuition); it seems to me that, given that cows have four stomachs, it follows that cows have more than one stomach (inferential appearance).
Note:ES DI APPARENZE RAZIONALI

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There is no rational basis for discriminating fundamentally among species of appearances. What makes sensory experiences a source of justification is the same thing that makes memories and intuitions a source of justification:
Note:IL PRIMATO DEI SENSI NN HA SENSO....MEMORIA E INTUIZIONE

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It is, for example, arbitrary to hold that sensory experiences may be presumed reliable but that intuitions should not be.
Note:ARBITRARIO

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phenomenal conservatism is not the thesis that all appearances are true.
Note:CAVEAT

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how things initially appear is the rational starting point:
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unless contrary evidence
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7.3   Synthetic a priori knowledge
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7.3.1   The question of the synthetic, a priori

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•Empirical knowledge is knowledge whose justification depends essentially on observations
Note:DEFINZIONE

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•A priori knowledge is knowledge that is not empirical.
Note:DEF 2

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•An analytic statement is, roughly, one whose denial is a contradiction.
Note:DEF 3

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synthetic statement is one that is not analytic, that is, its denial is consistent.
Note:DEF 4

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is there any synthetic, a priori knowledge?
Note:LA DOMANDA DEGLI ULTIMI 3 SeCOLI

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can there be a sentence whose denial is not contradictory, but nevertheless we can see it to be true in a way that does not depend on observation for its justification?
Note:TRADOTTO

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7.3.2   The answer
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No object is completely purple and also completely orange.
Note:ES 1...DIFETTO D IMMAGINAZIONE

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For any two moments in time, one is earlier than the other.
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it is better to be happy than to be miserable.
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If x is inside y, and y is inside z, then x is inside z.
Note:4

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The appearance of white is more similar to the appearance of yellow than to that of any other chromatic color.
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No action can cause an effect to occur before the action itself occurs.
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The number three exists.
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There could have been only seven planets in the solar system.
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The probability that either it is raining or it isn’t is 1.
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numbers, spatial properties, colors, ethical value, decision theory, time, causation, probability, possibility,
Note:TUTTO È COINVOLTO NEI SINT A PRIORI

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Take a famous epistemological thought experiment: suppose that I am a disembodied brain being kept alive by scientists in a vat of nutrients; my brain is being artificially stimulated to create the illusion of a physical world, and so on. In this case, all of my observations are false. Nevertheless, even if this scenario is true, I still know that no object is completely purple and completely orange.
Note:PERCHÈ PARLIAMO DI APRIORI?

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produce the definitions of the relevant terms in the statement, substitute the definitions for the terms they define, then, from the denial of the resulting statement, derive a sentence of the form ‘A and not A’.
Note:COSA FARE X DIMOSTRARE L ANALITICITÁ

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Take the sentence, ‘No object is completely purple and also completely orange.’ The project of showing this to be analytic is stymied at the first stage by the fact that there is no verbal definition of either ‘purple’ or ‘orange’. (There is only an ‘ostensive definition’, that is, these terms have to be explained by pointing to examples of the relevant colors.)
Note:IMPOSSIBILE NEL NS CASO

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‘Something is completely purple and also completely orange’, has the same logical form as ‘Something is completely furry and also completely happy’, which is not a contradiction.
Note:ANALOGIA

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7.3.3   How much synthetic a priori knowledge is there?
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they are willing to accept analytic a priori knowledge, but not synthetic a priori knowledge.
Note:L EMPIRISMO CADE O VINCE QUI... LA CONOSC ANALITICA È MNIMIZZATA COME NN SOSTANZIALE (SOLO LINGUISTICA) PER SALVARE LA LGICA

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What is the source of our synthetic a priori knowledge? The best answer to this is rational intuition, and the best explanation for our justification for relying on rational intuition is phenomenal conservatism.
Note:LA SORGENTE

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Rational intuition is not relevantly different from observation, reasoning, or memory for present purposes; all of these are simply different species of appearances,
Note:TUTTO APPARE

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we have good reason for seeking multiple justifications for a given conclusion; if it is possible to support a conclusion through both observation and intuition, this is preferable to supporting it through only one of these means.
Note:VALE TUTTO

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7.4   Metaphysical possibility
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7.4.1   The meaning of ‘possible’ and related terms
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a proposition is possible if it could be true
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A proposition is necessary if it could not be (and could not have been) false;
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A proposition is contingent if it is neither necessary nor impossible; that is, it could be true and could be false.
Note:DEF

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‘logical possibility’ and ‘metaphysical possibility’.
Note:LE DUE POSSIBILITÀ DI CUI CI CCUPIAMO

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A proposition is said to be logically possible provided only that it is not contradictory.
Note:DEF

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Metaphysical possibility is undefinable;
Note:CONCETTO VAGO MA INTUITIVO

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7.4.2   Metaphysical possibility is broader than physical possibility
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even things that conflict with the laws of nature may nonetheless be in some sense ‘possible’,
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laws of nature could have been different.
Note:INFATTI

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‘It could have turned out that we lived in a perfectly Newtonian
Note:ESEMPIO

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7.4.3   Metaphysical possibility as conceivability?
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To say that something is metaphysically possible is something close to saying that it is coherently conceivable.
Note:CONCEPIBILE

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conceivable’ just means ‘possible to conceive’,
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a proposition might be impossible for human beings to conceive solely because of some cognitive defect or limitation on our part.
Note:ALTRO PROBLEMA

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conceivability by an ideal observer,
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7.4.4   Metaphysical possibility and science fiction
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The notion of metaphysical possibility can also be illuminated by considering the limitations of fictional stories.
Note:UNA BUONA ANALOGIA: LA CRITICA DEI FILM DI FANTASCIENZA

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in Star Trek, people travel faster than the speed of light all the time.
Note:NESSUNO OBBOETTA...C È SOLO UNA FISICA DIFFERENTE

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aspects of stories are rejected by alert fans as ‘not making sense’. Time travel stories sometimes evoke this kind of criticism,
Note:REVERSIBILITA' DEL TEMPO...ESEMPIO CLASSICO DI IMPOSSIBILITÀ METAFISICA

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on the ‘branching timelines’ theory, when a person ‘goes back’ to some earlier time t, what actually happens is that a new branch of the universe is generated
Note:INVENZIONI X RENDERE CONCEPIBILE L NCONCEBIBILE

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there is a certain very broad sort of possibility that ought to constrain stories, such that if a story is impossible in this broad sense, it is thereby defective.
Note:IL PUNTO DI QS DISCUSSIONE....C È UN IMPOSSIBILITÀ CHE VA OLTRE LE LEGGI DELLA FISICA....IMPO METAFISICA

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critics are claiming that the time travel stories are metaphysically impossible.
Note:COSA FANNO I CRITICI DEI FILM DI FANTASCIENZA. ALTRI ESEMPIO: IL TELETRASPORTO A ZERO COINVILGIMENTO DELLA MATERIA

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7.4.5   Metaphysical possibility as a priori knowability?
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Metaphysical possibility is closely tied to the notion of a priori knowledge.
Note:SI NOTI CHE

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Thus, we know a priori that pentagons have five sides, and it is metaphysically necessary that pentagons have five sides; we know a priori that no object is completely purple and completely orange, and it is metaphysically necessary that no object is completely purple and completely orange.
Note:ESEMPI...VALE SIA X L ANALITICO CHE X IL SINTETICO

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7.4.6   Logical vs. metaphysical possibility
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The synthetic a priori truths are metaphysically necessary, yet their denial is not contradictory
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The synthetic, necessary, a priori truths that I have been defending are anathema to empiricists.
Note:RIPETIZIONE

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7.4.7   False signs of possibility: Analogies and mathematical systems
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No one should believe that a proposition is metaphysically possible merely because it is logically possible.
Note:PRIMA AVVERTENZA

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Mathematical systems in modern times are constrained by nothing other than mere consistency
Note:SECONDA AVVERTENZA...NEANCHE LA POSSIBILITÀ MATEMATICA GARANTISCE QUELLA METAFISICA

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the fact that one can develop a coherent mathematical system in which one talks about ‘infinite numbers’ such as ℵ0, does not show that any such numbers exist or could exist.
Note:ESEMPIO

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the argument by analogy: A is possible, and there is an analogy to be drawn between A and B; therefore, B is possible. This is usually fallacious, particularly when the analogy is based upon mere similarity of mathematical structure.
Note:ALTRO ARGOMENTO FALLACE...TERZO

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it is possible to travel in either direction along a spatial dimension, and there is an analogy to be drawn between the temporal dimension and a spatial dimension
Note:ESEMPIO

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7.5   Possibility and paradox
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I argue that most of the paradoxes of the infinite involve scenarios that are logically possible but metaphysically impossible.
Note:TESI

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I hold that there are many more necessary truths than are standardly recognized by philosophers, scientists, and mathematicians;
Note:LA VISIONE STANDARD È DOVUTA ALL EMPIRISMO

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I think there are many fewer possibilities than are commonly recognized.
Note:CONSEGUENZA...CHE RISOLVE I PARADOSSI

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For an empiricist, nothing can be ruled out a priori unless it is actually a contradiction
Note:GLI EMPIRISTI SI APPIATTIVANO SULLA MERA LOGICA SNOBBANDO L INCONCEPIBILE

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Even though empiricism is less popular than it once was, contemporary philosophy retains a significant resistance to synthetic necessary truths.
Note:PURTROPPO

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Suppose we want to know whether some proposition P is metaphysically necessary, impossible, or contingent. Many think that the presumption should be that P is contingent,
Note:BURDEN OF PROOF...UN PORTATO DELL EMPIRISMO

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If, however, metaphysical possibility is distinct from logical possibility, and if there are many things that are logically contingent yet metaphysically necessary or impossible, then it is unclear why there should be any such burden of proof.
Note:L'IMPOSSIBILITA' METAFISICA CI TOGLIE MOLTI IMBARAZZI... MOLTI PARADOSSI POSSONO ESSERE LIQUIDATI

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7.6   A realist view of mathematics
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the nature of mathematics.
Note:IL TEMA

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a body of a priori knowledge, some of it analytic and some synthetic.
Note:OPINIONE....COS È LA MATEMATICA....ANALITICA L INFERENZA...SINTETICA L ESISTENZA. ES: IL NUMERO 3 ESOSTE (GIUDIZIO SINTETICO A PRIORI)

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properties of and relations between certain kinds of universals.
Note:OGGETTO

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It is, however, necessary that there could be things exemplifying the property.
Note:IMMANENTISMO

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Arithmetic studies the properties and relationships of numbers, where numbers are properties that groups of things (whether physical or non-physical) can exemplify – for example, whenever there are two of anything, the property of twoness is exemplified.
Note:ESEMPIO...REALTA' CHE EMERGONO DA ESEMPI SEMPRE PRESENTI...IMMANENTISMO

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This view of mathematics (which is a form of mathematical Platonism) always seemed to me completely natural, so much so that I have never seriously entertained any other view. Nevertheless, the view is quite controversial in contemporary philosophy of mathematics.
Note:L'IMMANENTISMO MATEMATICO LA VIA PIU' NATURALE

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Some philosophers and mathematicians believe, instead, that mathematics is essentially a conventional symbol-manipulation game:
Note:CONCEZIONE ALTERNATIVA

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mathematicians invent a series of symbols,
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the whole system is pure invention,
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That view of mathematics, known as ‘formalism’, is motivated by empiricism. The empiricists of the twentieth century were loath to admit that there was any a priori knowledge of objective reality,
Note:FORMALISMO NOMINALISTA.... UN PORTATO DELL EMPIRISMO PER LIMITARE L APRIORISMO

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Some empiricists turned to the formalist philosophy of mathematics to rescue them from what would otherwise be a striking collection of counterexamples to their epistemological theory.
IMBARAZZO @@@@@@@@@@@@@@@

12 Some Paradoxes Mostly Resolved HL

12 Some Paradoxes Mostly Resolved
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12.1   The arithmetic of infinity
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from ‘∞+1 = ∞’, one can seemingly derive that 1=0;
Note:PUZZLE

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The expression ‘∞+1’ makes no sense because addition is an operation on numbers,
Note:SOLUZIONE FACILE

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The Cantorian approach, by contrast, holds that there are infinite numbers, but they simply obey different rules from all the other numbers.
Note:ALTERNATIVA CLASSICA

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12.2   The paradox of geometric points
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an extended region of space with some positive volume is supposed to be composed entirely out of parts (points) each of which has zero volume.
Note:PUZZLE 2...ASSOMIGLIA UN PO A ZENONE

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reject additivity of sizes for uncountable collections.
Note:SOLUZIONE TRADIZIONALE

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If a region is really built up from points, how is it that its magnitude is not explained by their magnitudes and number?

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NN C È UNA VERA SPIEGAZ

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this is what we must say to avoid paradox.
Note:L UNICA SPIEGAZ

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The simplest way to avoid these paradoxes is to reject the existence of size-zero parts of an object.
Note:COME EVITARE IL PARADOSSO

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there in fact are no such things as points; there are only positive-sized regions.
Note:ABOLIRE ABOLIRLA PUNTI

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12.3   Infinite sums
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the same (infinite collection of) numbers can seemingly have different sums, depending on the order in which the numbers are added.
Note:IL PARADOSSO

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we can obtain 0 if we add them like this: (1−1) + (2−2) + ... . But we can obtain ∞ if we add them like this: 1 + (−1+2) + (−2+3) + ... .
Note:ES PRATICO

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the correct sum sometimes depends upon the order in which the numbers are added.
Note:STANDARD VERSION

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The sum of an infinite series of numbers is defined to be the limit (if there is one) of the sequence
Note:NIENTE LIMITE NIENTE SOMMA

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it has the consequence that ‘sum’ has a different meaning
Note:CONSEGUENZE

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I agree in essence with the standard view of infinite sums,
Note:CONCLUS

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one cannot add up infinitely many numbers.
Note:IN SINTESI...LA SOMMA ESISTE MA NN RIUSCIAMO CALCOLARLA

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The real thesis is this: as a conceptual matter, addition is an operation defined on pairs of numbers.
Note:MA LA VERA TESI

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One can define an ‘infinite sum’ using limits, but then one is changing the meaning of ‘sum’.
Note:SOMME E LIMITI

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This explains why ‘infinite sums’ may violate the rules of ordinary addition, specifically, the commutative and associative laws
Note:ESSENDO COSE DIVERSE

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‘Infinite sums’ violate the rules for addition because they are not actually sums.
Note:CONCLUSIONE

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12.4   Galileo’s paradox
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in answer to the question ‘Which are more numerous: the perfect squares or the natural numbers?’, we can only answer that both are beyond number;
Note:RICRDA...L INFINITO NN È UN NUMERO...TUTTI I PARADOSSI CHE LO CONSIDERANO TALE SONO SMASCHERATI

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This thesis implies a rejection of the conclusion that is usually said to be established by Cantor’s famous Diagonalization Argument
Note:LA TESI DEL NON NUMERO

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the set of real numbers is larger than the set of natural numbers.
Note:LA TESI DELLA DIAGONALE....ASSURDA SE GLI NFINITI NN SONO NUMERI

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there is no one-to-one function from the natural numbers onto the real numbers,
Note:CIÒ CHE SI LIMITA A DIRE LA DIAGONALE

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it is not the same as the conclusion that the real numbers are more numerous than
Note:Ccccccvccc ALTRO ASSUNTO NECESSARIO > = ONE-TO-ONE

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there is a one-to-one function from a proper subset of the former onto the latter.
Note:RICORDIAMOCI PERÓ CHE ESISTE UNA ONE TO ONE TRA I MATURA I E UN SOTTOINSIEME DEI REALI

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the proper subset criterion (every set is greater than any proper subset of itself) – which is equally important.
Note:UN ALTRO CRITERIO PER IL GREATER THAN

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the best explanation for the fact that the two ‘criteria’ for greaterness often diverge in the case of infinite collections is that infinity is not a kind of number.
Note:IN QS CASO NN DIVERGONO MA SPESSO SÌ IL CHE RENDE PRUDENTE ASSUMERE CHE LA RELAZIONE <> NN ESISTA TRA GLI INSIEMI

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12.5   Hilbert’s hotel

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the completely occupied hotel with infinitely many rooms is able to accommodate one new guest by moving everyone to the next room down.
Note:PARADOSSO

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there will always be one guest temporarily out of a room.
Note:NN CI STANNO TUTTI

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‘Why suppose that the hotel guests relocate in sequence,
Note:OBIEZIONE

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Why not suppose that the hotel manager notifies all the guests of the new arrangement, using his PA system
Note:SINCRONIA

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The answer is that there cannot be such a PA system,
Note:RISPOSTA

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would require a signal that travels at infinite velocity.
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Velocity is a natural intensive magnitude,
Note:UN DATO CHE RENDE TUTTO IMPOSSIBILE

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now suppose that no communication was sent out. Instead, each of the guests, completely on his own, just decides
Note:SUPERIAMO L OSTACOLO

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This is metaphysically possible,
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the probability of such a coincidence happening is zero.
Note:PROBLEMA PLAUSIBILITÀ

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12.6   Gabriel’s horn
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in the real world, there is a limit to how thin a coat of paint can be (for example, it cannot be less than one molecule thick).
Note:NEL MONDO REALE UN SIMILE OGGETTO NN PUÓ ESISTERE...È DI X SÈ METAFISICAMENTE IMPOSSIBILE

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then the Horn could not in fact be painted with a finite amount of paint.
Note:QUINDI

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12.7   Smullyan’s infinite rod
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12.8   Zeno’s paradox
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1.To reach the ground, the ball must complete the series, ½, ¾, ⅞, ... . 2.This series is endless. 3.It is impossible to complete an endless series. 4.Therefore, it is impossible for the ball to reach the ground.
Note:RICORDIAMO L ARGOMENTO

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confusion between two meanings of ‘endless’ and two meanings of ‘complete’.
Note:CONFUSIONE

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no last member.
Note:INFINITO 1

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There is no time at which every member of S has    occurred.
Note:INFINITO2

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The last member of S has occurred.
Note:COMPLETO 1

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Every member of S has occurred.
Note:COMPLETO 2

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cannot be completed1

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INFINITO 1

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it is never completed2
Note:INFINITO 2

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Suppose I am going out of town for the week, and I get you to promise to feed all of my pets while I’m gone. If I have a turtle, then to keep your promise, you would have to feed the turtle. On the other hand, if I don’t have a turtle, then feeding my nonexistent turtle is not required. Similarly, if series S has a last member, then going through the last member is required
Note:ANALOGIA

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if S does not have a last member, then going through this nonexistent member is not required. The Zeno series has no last member, so reaching this nonexistent member is not required
Note:Cccccccc

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the series is endless1 but not endless2; it is never completed1 but it is completed2
Note:LA SERIE DI ZENONE

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12.8.2   The staccato run
Note:Ttttttttttttttt

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a person goes through an endless Zeno series, but the person pauses after each part of the journey.
Note:VARIAZIONE CON LO STACCATO

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this infinite series is impossible
Note:TESI

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this variation of the Zeno series, unlike the original version, requires infinite natural intensive magnitudes.
Note:METAFISICAMENTE IMPOSSIBILE

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average velocity remains constant
Note:PRIMA NOTAZIONE

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in each stage, you have to accelerate from speed zero up to your peak velocity, then decelerate back to speed zero
Note:ESIGENZA

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This means that your (average) acceleration must double with each succeeding stage.
Note:Cccccccccccc

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The runner’s body will thus need infinite material strength in order to avoid disintegration;
Note:Ccccccccc

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12.8.3   The short staccato run
Note:Tttttttttt

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12.9   The divided stick
Note:Tttttttt

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12.18   Comment: shallow and deep impossibilities 12.18.1   Solving paradoxes by appeal to impossibility
Note:Tttttttttttttt

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One way of solving such a paradox is to explain how the scenario is impossible.
Note:LA SOLUZIONE PIÙ COMUNE DEI PARADOSSI

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there are stronger and weaker kinds of impossibility.
Note:MA......

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some reasons for counting a scenario impossible would fail to resolve a paradox.
Note:PURTROPPO

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Suppose I could argue, completely convincingly, that (i) a budget of at least $200 trillion would be required to construct a lamp such as the one Thomson describes, and (ii) no one would ever be able to marshal so many resources for such a frivolous purpose.

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ES THOMPSON

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this obviously would not count as a solution to the paradox.
Note:Cccccccc

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Even physical impossibility may not be enough.
Note:ANCORA

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my argument turned on precise estimates of the value of Coulomb’s constant (the proportionality constant that relates electrostatic force to charge and distance).
Note:ES SEMPRE SULLA LAMPADA

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Suppose that if Coulomb’s constant were 10% larger than it actually is, then my argument would fail and the Thomson Lamp would be constructible; yet given the constant’s actual value, the Lamp is unconstructible.
Note:Cccccccc

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‘If Coulomb’s constant had been 10% larger, and the Thomson Lamp had been constructed, then what state would the lamp have been in at the end of the infinite series of switchings?’
Note:RISPOSTA OVVIA DELL OBBIETTORE

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if a certain device fails to exist due to some deep conceptual, logical, or metaphysical impossibility,
Note:DI COSA ABBIAMO BOSOGNO

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a metaphysically impossible scenario would have metaphysically impossible consequences.
Note:IN PARTICOLARE

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if the Thomson Lamp is impossible in a deep sense, we may think that there is no point to asking what would happen if the lamp were to exist, and that there need be no determinate answer to such a question.
Note:ESEMPIO

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12.18.2   An objection
Note:Tttttttt

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complain that my solutions to the paradoxes of infinity appeal to mere physical impossibilities, not metaphysical impossibilities,
Note:L OBIEZIONE

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have been at pains to emphasize that infinite natural intensive magnitudes are not ‘merely physically impossible’; they are metaphysically impossible, because they require there to exist a kind of number that does not exist.
Note:RISPOSTA.....INFINITI CHE NN SI COMPLETANO MA SI ESAURISCONO...VEDI ZENONE

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it is impossible to have more than four cats while also having fewer than three – this is deeply impossible because there is no number greater than four and less than three.
Note:ANALOGIA

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1.Scenario S requires magnitude M to take on an infinite value. 2.M is a natural intensive magnitude. 3.No natural intensive magnitude can assume an infinite value. 4.So S is (metaphysically) impossible. Having found S metaphysically impossible, I conclude that there is no need to concern ourselves with what would happen if S occurred.
Note:RIPETIAMO LA CLASSICA SOLUZIONE

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challenge whether (1) is metaphysically necessary.
Note:IL CRITICO POTREBBE OBIETTARE

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impossibility of infinite material strength
Note:ES...XCHÈ LA LAMPADA È IMPOSSIBILE

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One might object that it is only a contingent truth of our physics
Note:OBIEZIONE

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Perhaps there is some possible world with a different physics,
Note:Cccccccccc

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12.18.3   Deep physical impossibilities
Note:Ttttttttttt

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a sharper division between physical and metaphysical impossibilities
Note:E X RISPONDERE ENTRA IN GIOCO...

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Some ‘physical impossibilities’ are sufficiently deep to resolve paradoxes.
Note:TESI

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One can easily imagine the constant having a different value,
Note:X LA COSTANTE DI COLOMBO

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the switch need not have infinite material strength. It really is extremely unclear how we are supposed to imagine this working.
Note:CONTROFATTUALE IMPOSSIBILE DA IMMAGINARE

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an object withstands ever-increasing forces despite having only limited material strength?

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IMPOSSIBILE DA IMMAGINARE...CONTRADDOZ IONE

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it isn’t clear that such alternative physics are genuinely metaphysically possible.
Note:IL CONTROFATTUALE SEMBREREBBE...

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such as that nothing can be both completely red and completely blue,
Note:ANALOGIA

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someone wants us to imagine a radically alien physics,
Note:CHI CI CHIEDE QS SFORZO

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this would simply be to describe a different problem from the original one.
Note:SE UNO CI CHIEDE DI IMMAGINARE UNA FISICA ALIENA

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a.If I were to add a teaspoon of salt to this recipe, how would it taste? b.If I were to add a teaspoon of salt to this recipe, in an alternate possible world in which salt is a compound of plutonium and mercury and we are sea creatures who evolved living on kelp and plankton, how would it taste? Whatever might be said about (b), it is surely a different question from (a),
Note:ANALOGIA

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It is hardly an indictment of a theory that it fails to provide answers to hypothetical problems that have yet to be clearly posed.
Note:L OBIEZIONE SI INDEBOLISCE

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12.18.4   Generalizing
Note:Tttttttttt

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principles concerning binding energy are essential to the distinction between solid and non-solid materials
Note:ESEMPIO DI PRINCIPI TENUTI FERMI

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the relationship between resistivity and current flow is essential to the nature of electric circuits
Note:ALTRO ESEMPIO

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the Beer-Lambert Law is essential to the nature of transparency and opacity
Note:ESEMPIO

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To suppose these principles false is to suppose a fundamentally alien physics
Note:Cccccccc

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It is really not so clear that one could describe a coherent, metaphysically possible physics without these principles. Even if that would be possible, counterfactual questions about what would happen in an alternate world
CONCLUSIONE @@@@@@@@@@@@

IL PARADOSSO DI ZENONE RISOLTO

Riccardo Mariani

Adesso · 

IL PARADOSSO DI ZENONE RISOLTO

Achille, simbolo di rapidità, deve raggiungere la tartaruga, simbolo di lentezza. Achille corre dieci volte più svelto della tartaruga e le concede dieci metri di vantaggio. Achille corre quei dieci metri e la tartaruga percorre un metro; Achille percorre quel metro, la tartaruga percorre un decimetro; Achille percorre quel decimetro, la tartaruga percorre un centimetro; Achille percorre quel centimetro, la tartaruga percorre un millimetro; Achille percorre quel millimetro, la tartaruga percorre un decimo di millimetro, e così via all'infinito; di modo che Achille può correre per sempre senza raggiungerla.

Molti, tra l’insoddisfazione generale, considerano la storia di Achille come la dimostrazione che il movimento sia solo un’illusione (e con esso anche il tempo). Si va da Parmenide all’influente filosofo Emanuele Severino, ma anche l’originale scrittore da cui ho tratto la storiella: Jorge Louis Borges.

Tuttavia, ci sono modi più sensati di risolvere il paradosso delle serie infinite da completare. Di solito, infatti, si generano da una confusione tra due significati ben distinti di “infinito”: il primo considera tale una serie progressiva in cui non esiste l’ultimo membro (o ultimo evento), il secondo una serie in cui la realizzazione di tutti i membri (o eventi) richiederebbe un tempo eterno. La serie infinita nel primo senso non puo’ essere completata ma si esaurisce ugualmente poiché per esaurirsi non necessità affatto che di realizzi l’ultimo evento della serie. Questa analogia dovrebbe chiarire meglio il punto: se io ti chiedo durante la mia assenza di nutrire i cuccioli che ho in casa, tu li nutrirai tutti: se ho una tartaruga, nutrirai anche lei; ma se non ho una tartaruga tu, anche qualora non la nutrissi, avrai adempiuto ugualmente al tuo compito: cio’ che non c’è non puo’ realizzarsi. Allo stesso modo per adempiere una serie infinita non è necessario che si realizzi l’ultimo evento per il semplice motivo che in una serie di questo genere l’ultimo evento non esiste. La serie di frazioni che separa Achille dalla tartaruga, quindi, si esaurirà senza completarsi e Achille procederà a quel sorpasso che tutti noi vediamo e di cui non abbiamo affatto intenzione di dubitare (alla faccia di filosofi affascinanti ma improbabili quali Parmenide, Severino e Borges).

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