Ik kan iedereen het populair-wiskundig/filosofische boek "Infinity and the Mind" van Rudy Rucker aanraden. Er staan een paar dingen in waar ik het niet mee eens ben, maar over het algemeen is het erg goed en volgbaar. Kost ook maar iets van 25 euro ofzo, valt wel mee voor een wetenschappelijk boek. Van alles komt erin aan bod, van de oneindigheid van het Universum tot aan de wiskunde van 'large cardinals'. (Voor iedereen die dacht dat de limiet van de reeks 1, 2, 3, 4, ... goort was - dat is pas het begin van oneindig.
)
Voor de echte liefhebber post ik hier nogmaals de paper die ik zelf geschreven had viir het vak 'Infinity and the Mind' (inderdaad, zelfde titel als het boek, dat er dan ook voor gebruikt werd). Copyright ligt exclusief bij
mij, niet bij tweakers.net of welke andere persoon dan ook. (Voor een .doc versie met wat betere opmaak kan je mij mailen.)
Actualities, Possibilities and the Philosophy of the Infinite
A paper for CS Logica 2: Infinity and the Mind
By Victor Gijsbers
All human knowledge is either experience or mathematics.
- Friedrich Nietzsche, The Will to Power, book III
Overview
In this paper I wish to explore the idea that infinities in mathematics, though they cannot be conceived of as corresponding with any actuality, can be thought of as corresponding with possibilities. I will start by showing that mathematics must accept the correspondence-criterion for truth if it wishes to be fact instead of fiction; criticizing Ruckers ideas of infinite actualities, Ill try to show that these do not exist. Finitism can be seen as the mathematics of actualities. Non-finitist mathematics must be defined as a mathematics of possibilities if it wishes to have any explanatory power. But when seen as possibilities, mathematical objects cannot be said to exist in any reality unless constructed as mental image (intuitionism) or as string of signs (formalism); therefore, a Platonist view of mathematics is only possible with a mathematics that has no connections with reality, which makes Platonic mathematics the equal of fiction.
Introduction
Mathematicians, physicists and philosophers alike have discussed the infinite for centuries. Is infinite a clearly defined notion? Are there any physical entities that are infinite? Is our Universe infinitely large? Does time have a beginning or an end? Can thoughts be infinite? Is it meaningful to speak about infinite numbers? Can a calculus that uses infinite numbers be called true? In this paper I will be primarily concerned with the justification of non-finitist mathematics; Ill try to find out if there is a difference between that kind of mathematics and fiction.
Different schools of thought in the philosophy of mathematics have given different answers to the questions concerning mathematical infinities. In this paper I will look at a few of them: finitism, intuitionism, formalism and Platonism. Finitism claims that no infinities are allowable in mathematics. Intuitionism claims that infinities are allowable, but only if they are potentially constructible by human minds. Formalism claims that infinities are meaningful as long as they can be constructed as finite sequences of signs and have a definite role in a well-defined formal system. Platonism allows infinities while claiming that they are real, independent of construction.
The first question to be answered is whether any statement about the infinite can be said to be true. But what exactly do we mean with true?
Truth or Fiction?
There are three major ways of looking at truth. We can call a proposition P true when it corresponds to a fact in reality; we can call P true when it is consistent with a set of other propositions; and we can call P true when it is favorable to do so. I will not explore the latter, pragmatic, view here.
The consistency-criterion is much used. A statement like Lucifer and Beelzebub lay next to each other in the fiery lake can be said to be true in the context of Miltons Paradise Lost; it is consistent with everything we read there, and its negation is inconsistent with Miltons words. In the same way Lucifer is beloved by God would be false, since the poem shows Lucifer as a fallen angel whom is most certainly not beloved by God. The procedure for getting these truths is as follows: we have a certain number of axioms, the sentences of Paradise Lost. If either proposition P logically follows from these axioms and its negation does not, or if proposition P is consistent with the axioms and its negation is not, we call P true. It is possible to formalize this procedure to a mathematical system; taking a number of axioms, we can try to show that a certain proposition is either true or false.
However, the kind of truth we get this way is exactly the same kind of truth we get in the framework of fiction. I therefore think it is allowable to consider this kind of mathematics, that is, mathematics based on the coherence-criterion for truth, as a kind of fiction. Most people feel, as I do, that truths about fiction are not the truest kind of truths; statements that are really, objectively, true, are true by virtue of their correspondence with actual facts. If mathematics wishes to escape from the realm of fiction and enter the realm of fact, it must accept the correspondence-criterion for truth.
The correspondence-criterion states that a proposition P is true if and only if it corresponds to a fact. There are two kinds of facts: actualities, those things that really happen, and possibilities, those things that could conceivably happen.
Actualities
I define an actuality as anything existing as a complete and concrete, definite, fact (as distinguished from potential or possible). (See Thompson, 1999) Thus, my house is an actuality, whereas my house will collapse tomorrow is not an actuality but a possibility. Two classes of actualities can be distinguished: physical actualities (houses, trees, atoms, photons) and mental actualities (thoughts, desires). Though it is my conviction that all mental actualities are in fact physical actualities, in this discussion I will not assume that this is the case.
It is obviously possible to create a mathematics that is based on what is actual. Finite calculus, for instance, can be based on the physical properties of objects; if you put one object and another object together, youve got two objects; if you divide those two objects between two people, each has one object. Addition, multiplication and other such operations, when working on finite sets, can be seen as an abstraction from the realm of physical actualities.
Finite geometry is another example of mathematics based on physical actualities; in this case, were talking about an abstraction from such physical activities as measuring land. (See Husserl, 1936, for a defense of this view.)
What we are concerned with, however, is the question whether the manipulation of infinite sets, and of infinitesimals, can be founded on observations of physical or mental actualities. Rucker assumes that physical and mental infinities are possibly, if not provably, observable, but I wish to argue for the position that no actuality is infinite.
Infinite Actualities
Physical infinities can only exist if either space or time is infinite; physical infinitesimals can only exist if either space or time is (in principle) observable as a continuum. In the next few subsections Ill try to show that it is very likely that space and time are both finite and only discretely observable.
The Infinite
We do not know if there are any other Universes. There may be; there could be infinitely many of them. But these Universes would, by definition, be beyond our perception; we could never know anything about them. They cannot possibly influence anything in our Universe; so I conclude that their existence could not possibly influence my arguments, and Ill restrict myself to our own Universe.
One of the most certain theories in modern cosmology is the Big Bang theory; one would almost be tempted to paraphrase Dobzhansky and say: Nothing in cosmology makes sense except in the light of the Big Bang.1 This theory tells us that, somewhere between 10 and 15 thousand million years ago, our Universe began to exist. From a very tiny, very dense, very hot spot, it exploded and became the huge Universe we see today. We do not need to worry about the myriad of technical details of Big Bang Cosmology; the important facts are that a) the Universe is finitely old and b) the Universe is finitely large. It should be noted that the popular notion that the Universe started as a singularity is in fact false; at the first point we know about, the Universe was already the Planck-length across, some 1.5 * 10-33 meters.
We know, consequently, that neither time nor space is at this moment infinite. But what about the future? There are three major possibilities: a) the Universe will stop expanding at some finite time t, will start collapsing, and will end in a Big Crunch, b) the Universe will expand forever and ever, or c) the expansion of the Universe will become slower and slower, and will asymptotically go to zero. Current astronomical observations have made it very likely that c) is the case: in the limit to infinity, the Universe will stop expanding, and will become static. But doesnt this mean that there are indeed infinite actualities? Doesnt it mean that physical objects could exist forever?
No, not really. For time doesnt mean anything, cannot be physically measured, unless something happens. However, before infinity is reached, things will stop happening. The entropy of our Universe can never decrease; this is the Second Law of thermodynamics. When the Universe starts expanding at a slower and slower rate, the maximum amount of entropy in the Universe will stop increasing. In a finite time, a state will be reached where the entropy of the Universe is at its maximum; nothing will happen anymore. The total amount of energy in the Universe is fixed; as the size goes to infinity, the amount of energy per volume goes to zero. In the limit there will be only single particles, not interacting anymore, in a Universe without change. Radioactivity will no longer exist, for all radioactive substances will already have decayed. All protons will have decayed. Only stable particles will populate an otherwise dead Universe; not even black holes will exist anymore, for they have evaporated. In this Universe, the passage of time no longer has a meaning; one moment will be indistinguishable from the next; therefore, they will be the same. Distance stops being a meaningful concept, for all forces are zero. In the limit of infinite time, time stops being meaningful. No actualities can be said to exist forever.
The Infinitesimal
Yet what about the infinitesimal? Are time and space not continuous? No; quantum physics shows that all phenomena are in fact discrete phenomena. Any physical system evolves in time with quantum jumps, discrete, instantaneous changes in the wave function that describes the system. This was predicted theoretically when quantum mechanics was first formalized, and it has been verified experimentally.
A second result of quantum mechanics is that there is an inaccuracy in every measurement. This is not just an experimental limitation, but a fundamental fact about reality. Neither space nor time can ever be measured to an infinite precision; not by us, and not by any other physical system. It therefore seems rather meaningless to claim that objects in space-time are well defined on a continuum of points.
Though it might in theory be possible for an object to be on a continuum of points, no actuality is ever well defined on a continuum of points. The space-time continuum, therefore, is only a continuum of possible locations for actualities, and not a continuum of actualities themselves.
The last possibility of an infinity of actualities seems to lie in an infinite cascade of smaller and smaller particles; if any particle consists of smaller particles, there must be an infinity of particles. There are two ways to criticize this assumption: the first is that particles such as quarks cannot exist independently; they always must form bigger particles with other quarks, they just dont fly around by themselves. This makes it very dubious whether quarks, let alone smaller particles, can be thought of as actualities. The second critique is that there is no scientific reason at all to believe that there will always be smaller particles than the ones known. This is an empty assumption, and we do not need to bother with it until it is in some way (experimental or theoretical) justified.
Infinite Thoughts
Moving from the physical world to the mental world, we will now look at the existence of infinite thoughts, desires or other mental phenomena. It should be noted that a possible thought is not an actuality; only a thought that is actually thought is an actuality. Actual thoughts depend on actual minds. Can finite minds think infinite thoughts? An infinite thought, if it is to be entirely definite, has an infinite complexity. It is of course possible to think about a sequence such as 1, 2, 3, ... ω, but this sequence is not entirely definite, as evidenced by the appearance of three dots between the 3 and the ω. Since an actuality must, by definition, be definite, an infinite actual thought must be infinitely complex; if it is thought of in a finite way it loses part of its definiteness.
If a finite mind can think infinite thoughts, it must think not entirely definite versions of them that are not infinitely complex; for to think an infinitely complex thought, ones mind must be infinitely complex itself, and thus infinite. The existence of infinite actual thought is therefore dependent on the existence of infinite minds. Do infinite minds exist?
Not as far as we know. The human mind seems to be finite. A possibility is of course God; by definition She must have an infinite mind, if She exists. A detailed analysis of the concept of God requires much more space than I can provide here, but it seems pretty clear that there is currently no known way to prove the existence of god by any logical argument. An appeal to faith, though it might have some significance in mystical thought, has no use here: of course its possible to prove that infinite actualities (in this case Gods thoughts) exist if you postulate the existence of an infinite actuality (God); but that is not allowable when we try to find out, scientifically, whether infinite actualities do or do not exist.
I conclude that there is no reason to accept the existence of any infinite actualities, physical or mental. There might be, but we do not know for certain, and we do not think it likely; so well be prudent and assume that there are no infinite actualities.
Finitism
What does this mean for mathematics? As we have seen in the section Truth or Fiction? mathematics must base itself on some aspect of the real world if it wishes to be more than fiction. A mathematician might, therefore, wish to base mathematics on actualities, physical or mental. But as no infinite actualities exist, this mathematics cannot feature any infinities either. Finitist mathematics, therefore, can be seen as mathematics based on actualities. The finitist tries to describe what is real, what is actual, and for this he needs no infinities. In fact, any infinities will arouse suspicion in him; what would you need those for?
I conclude that finitism is the only reasonable mathematics for a mathematician who wishes to limit himself to actualities! Finitist mathematics should in principle be enough for physics - or should it? Physical theories do not limit themselves to actualities. It is for instance very physical to claim that sugar is soluble in water. This is true for any bit of sugar; regardless of whether it ever comes in contact with water. Physics, therefore, does not only talk about actualities. It also talks about possibilities: If situation A were to arise, B would happen. In the next section well see that possibilities can in fact be infinite. Finitist mathematics might be too limited even for physics.
We may not wish to limit mathematics itself to the finite. Large parts of existing mathematics work with infinities, and we would like to find something in the true world that these mathematics might be said to describe. We do want, after all, that mathematics is different from fiction. And in fact, infinities can be found in reality; not as actualities, but as possibilities.
Possibilities and the Infinite
Thompson (1999) argues that mathematics is intimately connected with the question of `possibilities'. According to him, this is often ignored due to the popularity of extensional semantics, which claims that the meaning of any mathematical property is precisely the set of all objects which have that property. However, extensional semantics has one problem: it assumes from the outset that the things it talks about are eternal; that they cannot be created or destroyed. In the real world, however, those kinds of changes do happen.
Suppose for instance that Im going to write a paper about ants tomorrow, and that you too are going to write a paper about ants tomorrow. From this I conclude that together we will have two papers about ants tomorrow. But if I strictly adhered to extensional semantics, I would face a problem. Neither the set of my papers about ants, nor the set of your papers about ants exists presently. (Or rather, they are both the empty set.) How could I use set theory on sets that do not actually exist? How could I add non-existent sets, and claim to know what is the result of that addition?
There is a way around this problem. We must interpret the statement
If X and Y are two disjoint finite sets, then the number of elements in their union, , is the sum of the number of elements in X and in Y,
as a statement about possible sets; whenever X and Y are two disjoint finite sets, then the number of elements in their union will be the sum of the number of elements in X and in Y. A mathematical theory that wishes to be able to describe reality must be able to talk about possibilities, and not just about what already exists.
Now that we know that mathematics and possibilities are closely linked, we must find out whether there are any infinite possibilities. It certainly seems there are. Though it cannot be measured even in principle, it seems at least possible that a particle can be on a continuum of points; that, in other words, there are infinitely many possible places for a particle to be in an finite part of space. If we agree to this, many quantities are seen to have infinitely many possible values: velocity, kinetic energy, potential energy, etcetera.
Infinities are, therefore, needed to describe real possibilities. A mathematics that uses infinities can indeed be more than fiction; it can take its axioms from the real world as easily as finitist mathematics. What influence has this conclusion on the three most important philosophical positions in mathematics, namely intuitionism, formalism and Platonism?
Intuitionism
Time, as a moving process, is intuited by the human mind. When it is seen that there is a before and an after, the mind has constructed a two-ity; from this fundamental two-ity all the mathematics of natural numbers can be formed. There is nothing wrong with infinities, as long as we see that they are potential, not actual. This is, very much simplified, the way that Brouwer though about mathematics.
Whether or not time is the basis for mathematical intuition, and whether or not an intuition can constitute knowledge, is beyond the scope of this paper. What concerns us here is the intuitionists idea about infinity. He claims that infinities are not actual, but potential: they are not constructed, but they might be constructed. We can certainly agree with the first part, there are no actual infinities. But whether or not infinities can be constructed seems more dubious. I think it is justified to assume that Brouwer means that infinities can potentially become actualities: they are not constructed yet, but might be constructed at an infinitely removed later time. If, as I argued, no actual infinities can exist even in the limit to infinite time, this view is false. Infinities are not actual, but they are not potential either: they could never be constructed; they are sets of possible actualities (or sets of sets of possible actualities), not possible sets of actualities.
An intuitionist can easily change his position to one compatible with my conclusions on mathematical infinities; he has to accept possibilities, not merely actualities, as things which can be intuited. But if he does not do so, and still talks about infinities, he is just telling fiction.
Formalism
A mathematical theory is based on a number of axioms, and rules to manipulate those axioms. Basically, mathematics is the theory of sign-conventions; but one can create axiomatic systems that correspond with reality. This is, simplified, the philosophy of formalism.
Formalism does not really need a justification; as the theory of manipulating strings of signs, it makes no claim to truth or reality. However, the question is whether formalistic mathematics could ever describe reality; whether it could make a claim to be true. The answer to this question is, I think, affirmative. Since mathematics can be based on real possibilities, it is possible to create a set of axioms that, when interpreted, are true about these possibilities. One can then use these axioms as the basis for a mathematical theory. When interpreted, the results of this theory will again correspond to real possibilities.
As long as the one who uses formalistic mathematics is careful not to interpret his results as statements about actualities, but as statements about possibilities, there is no problem with his approach to mathematics or his claim for truth.
Platonism
Mathematics is about real things; not physical actualities, but about objects in an eternal and unchanging realm of ideals. Mathematics is not a process of invention, but a process of discovering these eternal truths. This is, again simplified, the basic supposition of Platonism (in mathematics).
However, we saw in the section on possibilities that a mathematics that limits itself to existing sets is too limited to be able to describe our changing reality. Under the assumption that Platonism is correct, we cannot interpret a mathematical statement as a statement about possibilities; it must be a statement about actualities. Thus its axioms could be justified by observing reality; but only in the case of finite mathematics. Infinite actualities do not exist in our physical or mental observation, so any statement about infinities, if it is to correspond with something in the observable world, must be about possibilities and not about actualities.
But Platonism can only talk about actualities, for it talks about actual objects; and it talks about infinities too. From this we must conclude that Platonist mathematics cannot be justified by observing reality; there is no way to show that it is any different than fiction. Platonism has no connections with reality; therefore it is fiction.
Conclusion
Intuitionism, when properly formulated, and formalism, when properly interpreted, are both philosophies of mathematics than can rightfully claim that their subject matter is something different from fiction. They can talk about infinities because these statements can be interpreted as statements about possibilities. Finitism is the mathematics of actualities, and as such terribly limited, if well-justified. Platonism, on the other hand, severs all connections with reality. It cannot claim that its mathematics is any different from fiction.
Notes
1 Theodosius Dobzhansky once made the much-quoted statement: "Nothing in biology makes sense except in the light of evolution."
Literature
Bransden, B. H. & Joachain, C. J., Quantum Mechanics, 2nd edition, 2000, Prentice Hall
Danzig, D. van, Is a finite number?
Dieks, D. Inleiding in de grondslagen van de natuurkunde: struktuur en betekenis van fysische theorieën, 1997, Utrecht University
Hawking, S., A Brief History of Time, 1988, Bantam Books
Husserl, Vom Ursprung der Geometrie,1936
Milton, J., Paradise Lost, 1667
Nietzsche, F., The Will to Power, 1968, Vintage, New York (Translated from German)
Oppy, G., Gödelian Ontological Arguments, 1996, Analysis 56
Rucker, R., Infinity and the Mind, 1995, Princeton Science Library
Sakurai, J. J., Quantum Mechanics, Revised Edition, 1994, Addison-Wesley Publishing Company
Thompson, I., Philosophy of Nature and Quantum Reality, 1999, (
http://www.ph.surrey.ac.uk/~phs1it/papers/pnb/node2.html
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decimalen. Dan moet die leraar van jou wel heeeel erg klein hebben geschreven om dat op 5 a4-tjes te passen ! :strip_exif()/u/34942/daniel.gif?f=community)
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