Is Science without Religion really Lame

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Drumz wrote : ``1. Science has you believe BLINDLY in concepts you have no way of seeing, measuring or even conceptualizing (whats that remind u of?). The ``number`` ZERO for one.``

Zero is measurable and conceptualizable (except to people of IQ Zero). Being able to express something mathematically means one can conceptualize it, and that too, with greater precision than many other ideas (e.g., ``God`` or ``good``). Anyway, ``believing blindly in what you have no way of seeing`` seems oxymoronic, I mean, is there any other way of believing in something that you cannot see? I cannot see air or space, btw, but I believe I have grounds to believe in them :)

Dear Readers:
I had alluded to a famous comment by Einstein in the paper: God does not play dice with the world. I had also quoted Chandrasekhar`s quip to Einstein`s cliche. I give hereunder two more by Niels Bohr and Stephen Hawking respectively.

Niels Bohr had responded, ``Albert, stop telling God what He can do.`` And Hawking commented much later on, ``God not only plays dice. He sometimes throws them where they can`t be seen.`` It seems Niels Bohr`s comment was a tete-a-tete.

Mohammad Gill

DRUMZ #144
You have failed to convince me that one cannot own zero apples or negative apples.

If you own 120.01 dollars, for instance, you can own one hundred dollars note + two 10 dollar notes + ZERO one dollar notes + ZERO 10 cent coins + one 1 cent coin.

Or you can own ZERO 100-dollar notes, twelve 10-dollar notes, ZERO 1-dollar notes etc.

When you cannot find a quarter in your purse for the parking meter, you understand what is zero. The homeless guys who line up at soup kitchens have zero money. Nothing uneffable about their predicaments.

I also donto see why absence of motion doesnot correspond to zero velocity. Velocity is a concept defined by man as a characteristic of motion. This man-defined concept corresponds nicely with certain characteristics of motion of real life objects.

I also have no problem with the real world representation of infinity. For all practical purposes, it is a very large number.

At other times, the concept of truly infinite infinity sometimes facilitates closed form solutions for mathematical expressions, solutions which are logically consistent at non-infinity points. In other cases, the concept of infinity raises a red flag about limitations of applying certain mathematical representations to real world things. In short, I have had no quarrel with infinity as an engineer.

If I were a physicist delving into the unknown, I might have to be more circumspect about infinity. But thats the fun of mathematical systems and seeing how far they get you.

``Because this is chowk and u guys spend years on end having useless sophmoric discussions about who has the coolest country (re U, urstruly, jay etc) , i did make it a point in almost EVERY post to recap my points and keep stating my thesis. ``

That`s the second time at least that you have gotten personal with me though we are discussing math. I could make a lot of personal remarks about you too if I wish, but I thought my opinion of you was irrelevant to our understanding each other`s points in this discussion.

It seems you are not really interested in an honest discussion about math, you are more interested in some sort of payback. So goodbye.

DRUMZ #133

{{Your other points are discussed above except for the chess one (this IZ a Queens Gambit)}}

DRUMZ, I was not interested in whether you could drop names of chess openings. There is a good reason why I asked you in #126 about whether you knew the rules of chess, and also about the Peano axioms (which I mentioned more than once). Had you been somewhat knowledgeable and/or insightful you might have understood the connection. It appears that you did not.

For those who know chess, or at least the rules, I was going to ask: what is the quiddity, the quintessence of the concept of chess ‘knight’, of the other chess pieces, of the chessboard?

Now chess is conventionally played on a physical board, ruled in squares, with pieces of ivory and ebony (or coloured plastic, in these degenerate times). Chess moves are conventionally carried out by actually moving the pieces between the board squares. Thus a game can be represented in terms of some actual events in the material world.

Does it then follow that a chess game only ‘exists’ to the extent that it unfolds in the material world? Is the ‘nature’ of the knight determined by its size, shape, colour, heft, the material of which it is made?

I assert that that is not the case, and in fact size, shape, weight, chemical composition etc. have nothing to do with the properties of the concept ‘knight’. Its properties are determined entirely by its definition in the abstract formal system we call ‘chess’, which describes what the starting state is, which moves are ‘legal’, etc. We ‘construct’ chess games by following the rules of this formal system. We can ask questions about this system such as: Can the knight ever leave and then return to its starting point in an odd number of moves? Can the knight visit all sixty-four squares of the chessboard and return to its starting point without having visited any other square twice? What is the maximum number of queens (of both colours) which can be on the board at the same time during any chess game (however contrived)? Does a specific position have a forced win for White (or Black)? Etc. The answers to these questions (some trivial, others unknown or not so trivial) are all determined by the made-up rules of the game.

Now chess (a chess position, chess game, etc.) has many representations. First there are ‘physical’ representations via actual chessboard and chess pieces, or diagrams printed in books. I claim that the answer to questions about the game ‘chess’ does not in any way depend on physical representations. Physical representations serve as crutches for human beings to use in visualizing and reasoning about board positions, but they are not essential to such activities and have no bearing on the truth or falsity of any statement made about chess.

I hope this take seems ‘reasonable’ to some people – I don’t know about DRUMZ or people with concrete-bound mentalities.

In addition to physical representations, one might also look at symbolic representations – discrete encodings – of a chess position, of chess moves, of chess games. The best known ones are called descriptive, algebraic, figurine algebraic – they are of course not the only possible ones. They are perfectly adequate for reasoning about chess. The essence of chess – the rules, the positions, the game – is what is common to all symbolic encodings.

It may also be relevant to ask whether there is anything ‘ineffable’ or ‘impossible to conceptualize’ about chess, and the chess knight, defined as a formal/symbolic system. I hope not – though DRUMZ and his ilk may not agree going by the history of this board.

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Now it is possible to treat the natural numbers (i.e. 1, 2, 3, .. ) in just the same way, i.e. to define them as ‘pieces’ in a game, elements of a formal system. This same approach may then be extended to include the rationals, 0, negative numbers, the reals.

The standard axiomatisation of the natural numbers (i.e. rule set for constructing the natural numbers and reasoning about them) was given by Giuseppe Peano and is known as Peano’s Axioms. It constructs the set of natural numbers using just the number 1 and a single function called ‘successor’. It does depend on the concepts of ‘set’ and ‘function’ – which may in turn be further axiomatized to an extent.

Given the 5 Peano axioms, it is possible to prove ‘familiar’ properties of the natural numbers – such as the commutativity and associativity of addition and multiplication, the distributivity of multiplication over addition, etc. Proof only requires that you follow the ‘Laws of the Game’ given in the Peano axioms.

Positive ‘fractions’ / ‘rational numbers’ may then be defined, and similar rules proved about them. Similarly with the positive reals (usually defined as so-called ‘Dedekind cuts’). Finally, the element 0, and negative numbers, may be added to the set of real numbers and more properties proved.

Note that the addition of ‘zero’ is equivalent to the introduction of a new piece in the game – together with rules of play governing that piece, the way it interacts with other pieces.

Please note very carefully that the properties of ‘zero’ as introduced this way do not depend in any way on some pompous inconceivable ‘ineffable’ notion of ‘NOTHINGNESS’, ‘No Thing’, etc. Zero - apart from a few singular mathematical properties - such as being its own additive inverse and the additive identity - is not different in its intrinsic character from any other piece in the game. Nor does understanding how the zero behaves require any great feat of INTROSPECTION.

Let me mention at this point that – just like chess - the number system can be symbolically represented in more than one way. For instance, one might use decimal, binary, hexadecimal, balanced ternary representations of integers, reals. What is interesting is that one and the same number may have more than one representation in an particular encoding – the encoding is then called redundant. For instance – do 0.4999999… and 0.50000000… (in the usual decimal representation) denote the same real number? They do – a convincing explanation follows from the ‘cut’ definition of real numbers.

We can also carry out for the numbers – defined as a formal system – a process in reverse to that which I carried out for chess above. For the case of chess, I went from the physical representation to the symbolic or abstract. For numbers, one can go from the abstract to a physical model. Thus natural numbers can be represented as glass beads or knots on a string, tied down at one end, and a suitable ‘Glass Bead Game’ defined to mimic the properties of the number line. This could be extended to negative integers and zero (the bead in the center of a string extending in two directions. Modulo arithmetic of course has the obvious physical model of a clock face. But the properties of the number system defined via Peano and extensions in no way depends on the existence of physical models.

I understand that DRUMZ might find all this difficult to OUTERstand. Let him not instantly fall into his usual habit of presuming that the fault lies with someone else’s presumed stupidity rather than his own intellectual shortcomings.

There is an old but still excellent axiomatic development of the number system – ‘Foundations of Analysis’ by Edmund Landau. It should be available in any university library. In a little less than a hundred pages, Landau shows how to rigorously define the familiar notions of ‘elementary’ mathematics - integers, fractions, real numbers, starting with the Peano Axioms. It is a worthwhile experience for anyone who doesn’t realize it can be done, to cure them of their notions about the number zero being an ‘ineffable’ quantity and other such inanities.

Incidentally, Landau’s book doesn’t contain a single diagram or picture. Pictures are an aid to visualization, imagination, but are not necessary to establish the truth of a proposition.

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More to come …

am Saying That The INVENTED system of mathematics contains variables which do not have representations in the physical world. SIMPLE.
This is known as the Representative theory of Reality-
This is a discreditied, and old fashioned view of our knowledge- I would suggest you stop laughing at the problem of Realism in sciences and mathematics and better get into the issues in it. You have to understand the fact that most of the entities- including the particles like gravitons etc are merely postulates-
Only religion creates permanent entities like God and souls- which does not explain any thing that human beings experience on this earth.

#145 by DRUMZ on January 28, 2004 8:48pm PT
.....and im debating my WEAKEST subject...

Boy! you are going to turn this discussion into a turning point of your life by getting introduced to some of the best topics of research in mathematics- and as an additional advantage, you might top your class with honours after carefully pondering over our discussions.

DRUMZ #137

{{All challengers on this thread have officially been DECAPITATED}}

Not so fast with those unilateral declarations of victory.

{{im debating my WEAKEST subject...}}

The saddest part is that you`re utterly ignorant about the full extent of your ignorance.

More to come...

..............REDLINE..............

All challengers on this thread have officially been DECAPITATED and THIS ladies and gentlemen is how u debate the bullsh1t outta someone. 5 against one and im debating my WEAKEST subject...

And No This duz not make me ``intelligent.`` This simply EXPOSES the chowk intelligentia for what they truely are...

Pseudo intellectual one dimensional half wits who`se ONLY talent is to spend countless time over the internet arguing about how Cool their country is and how inferior everyone else`s country is...

And here 5 of u were arguing a point which u simply BLINDLY accepted from ur masters. No one on this site ever THINKS about something before accepting it.

All U guys do is praise the very countries and religions u morons were born into.....! Now run along and dont speak a werd about this thread to anyone.

Sadna: Because this is chowk and u guys spend years on end having useless sophmoric discussions about who has the coolest country (re U, urstruly, jay etc) , i did make it a point in almost EVERY post to recap my points and keep stating my thesis.

I am Saying That The INVENTED system of mathematics contains variables which do not have representations in the physical world. SIMPLE.

For the billionth time.. It IS NOT POSSIBLE To OWN (I feel like Simon from american idol) to OWN (does anyone know what own means??????????) Something that DOES NOT EXIST.

Im a strict Logician incase u havent noticed. In the REAL WORLD, One may Own ONE Apple or NOT OWN one apple. But one CANNOT own NO Apple. There IZ a difference clearly..... To OWN or ``to Have`` INVARIABLY REQUIRES a Thing/Idea etc. It is IMPOSSIBLE to OWN something that does not exist. ``No apple`` is an apple that DOES NOT EXIST. If I was to give the apple to u, the I would ``NOT OWN`` the apple, and U would ``OWN IT``<-BUT THE APPLE WOULD STILL EXIST!!!!!!!!!!!!!!!!

If we must represent an individual not owning an apple, we say ``X does NOT OWN an apple.`` In the example (I dono why im explaining this in SOOO Much detial) There exists a Person, an agreement over ownership and a THING.

A THING as opposed to a NO-THING (ie the NO APPLE). Iz this making any sense to anyone??? One cannot talk about ownership about an item which cannot be owned. The Apple can be owned, but the ABSENSE of the Apple CANNOT BE OWNED. It ill ILLOGICAL.

Moving right along. ``Walking at the SPEED of zero.`` Again LOGIC. To WALK INHERENTLY IMPLIES MOTION. Motion cannot occur at the speed of the ``absense of motion - ZERO-... Ur example is illogical.

Also, One CANNOT own -2 apples. In REALITY (THE REAL WORLD!!!!) to OWN means to Possess. U cannot possess something that does not exist. -2 apples cannot be in anyones possession.......... U also cannot possess something that u do not possess.....!!!

Another recap: The negative signs may be ``useful`` as u say (somethin i never questioned) but they cannot be translated to the real world.

Now why is it that EVERYONE LEAVES OUT THE INFINITY EXAMPLE???????????? I am certain it will B much more easier to grasp for u guys then the zero. Now Sadna, Conceptualise INFINITI.

Ballu: Good links. Post 138 was the most accurate IMO...

Existence theorem
From Wikipedia, the free encyclopedia.

In mathematics, an existence theorem is a theorem with a statement beginning `there exist(s) ..`, or more generally `for all x, y, ... there exist(s) ...`. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not do so, as usually stated. For example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification is then hidden in definitions.

A controversy that is now quite old (about a century) concerns the issue of pure existence theorems, and the related accusation that by admitting them mathematics was betraying its responsibilities to concrete applicability (see nonconstructive proof). The point from the mathematical side was always that abstract methods are far-reaching, in a way that numerical analysis cannot be. An existence theorem can be called pure if the proof given of it doesn`t also indicate a construction of whatever kind of object the existence is asserted. But to speak in that way violates the standard way mathematical theorems are encapsulated: they may be applied without knowledge of the proof; and indeed if that`s not the case the statement is faulty.

From the other direction there has been considerable clarification of what constructive mathematics might be; without the emergence of a `master theory`. For example according to Bishop`s definitions the continuity of a function (such as sin(x)) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a promise that can always be kept. One could get another explanation from type theory, in which a proof of an existential statement can come only from a term (which we can see as the computational content).

DRUMZ
The reason I didnot reply is I donot see quite what are you arguing about.

``Take two apples in your right hand and no apples in your left.`` The two apples still exist, they have changed locations. One cannot HAVE a ZERO APPLE (you cannot Have or OWN that which is NOTHING). Its illogical. I can say Rite now I have zero universes in my pocket, zero countries are floating over my head.... A ``negative`` something cannot be quantified and thus is illogical... ``

One can own zero apples. One can own zero cars. You can certainly have zero universes in your pocket. One may want to REPRESENT zero apples. Zero apples as opposed to two apples is quite logical as a concept, IMO.

Negative numbers are not illogical. As you say, if you owe a man two apples, you own -2 apples. And for example, when you seek to represent directions along with quantities.

A man can walk due east at a speed of -2 km/hr, meaning that he is walking due west at the speed of 2 km/hr. If he stands still, he is walking at a speed of ZERO km/hr. What is so undefined about a man standing still? Nothing. It is understood that he is travelling at zero speed wrt the ground/surface of Earth, NOT at zero speed wrt the surface of some far off star say Alpha Centuri (which is moving at some humongous number of km/hr relative to Earth).

Negative numbers are useful to describe voltages, temperatures(ie where zero is taken as a given reference point). And kilometers and hours are man-made `units`, while voltages and temperatures themselves are concepts as are `due east `and `due west`.

It has been explained already many times on this thread that mathematics/number systems are languages and tools with defined rules/axioms. So I am puzzled when you assert that this is uniquely your viewpoint.

The issue of relating the mathematical system to/using it to describe physical phenomena and associated philosophical notions about these rules/axioms has also been discussed. I donot see that anything you say invalidates these arguments.

The final post on the indespensibility argument

Indispensability Arguments in the Philosophy of Mathematics
One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It`s not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.
From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine (1976; 1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called ``intellectual dishonesty`` (Putnam 1979b, p. 347). Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism. There are other indispensability arguments,[1] but this one is by far the most influential, and so in what follows I`ll concentrate on it.

1. Spelling Out the Quine-Putnam Indispensability Argument
2. What is it to be Indispensable?
3. Naturalism and Holism
4. Objections
5. Conclusion
Bibliography
Other Internet Resources
Related Entries

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1. Spelling Out the Quine-Putnam Indispensability Argument
The Quine-Putnam indispensability argument has attracted a great deal of attention, in part because many see it as the best argument for mathematical realism (or platonism). Thus anti-realists about mathematical entities (or nominalists) need to identify where the Quine-Putnam argument goes wrong. Many platonists, on the other hand, rely very heavily on this argument to justify their belief in mathematical entities. The argument places nominalists who wish to be realist about other theoretical entities of science (quarks, electrons, black holes and such) in a particularly difficult position. For typically they accept something quite like the Quine-Putnam argument[2]) as justification for realism about quarks and black holes. (This is what Quine (1980b, p. 45) calls holding a ``double standard`` with regard to ontology.)
For future reference I`ll state the Quine-Putnam indispensability argument in the following explicit form:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

Thus formulated, the argument is valid. This forces the focus onto the two premises. In particular, a couple of important questions naturally arise. The first concerns how we are to understand the claim that mathematics is indispensable. I address this in the next section. The second question concerns the first premise. It is nowhere near as self-evident as the second and it clearly needs some defense. I`ll discuss its defense in the following section. I`ll then present some of the more important objections to the argument, before considering the Quine-Putnam argument`s role in the larger scheme of things - where it stands in relation to other influential arguments for and against mathematical realism.
2. What is it to be Indispensable?
The question of how we should understand ‘indispensability’ in the present context is crucial to the Quine-Putnam argument, and yet it has received surprisingly little attention. Quine actually speaks in terms of the entities quantified over in the canonical form of our best scientific theories rather than indispensability. Still, the debate continues in terms of indispensability, so we would be well served to clarify this term.
The first thing to note is that ‘dispensability’ is not the same as ‘eliminability’. If this were not so, every entity would be dispensable (due to a theorem of Craig).[3] What we require for an entity to be ‘dispensable’ is for it to be eliminable and that the theory resulting from the entity`s elimination be an attractive theory. (Perhaps, even stronger, we require that the resulting theory be more attractive than the original.) We will need to spell out what counts as an attractive theory but for this we can appeal to the standard desiderata for good scientific theories: empirical success; unificatory power; simplicity; explanatory power; fertility and so on. Of course there will be debate over what desiderata are appropriate and over their relative weightings, but such issues need to be addressed and resolved independently of issues of indispensability. (See Burgess (1983) and Colyvan (1999b) for more on these issues.)

These issues naturally prompt the question of how much mathematics is indispensable (and hence how much mathematics carries ontological commitment). It seems that the indispensability argument only justifies belief in enough mathematics to serve the needs of science. Thus we find Putnam speaking of ``the set theoretic ‘needs’ of physics`` (Putnam 1979b, p. 346) and Quine claiming that the higher reaches of set theory are ``mathematical recreation ... without ontological rights`` (Quine 1986, p. 400) since they do not find physical applications. One could take a less restrictive line and claim that the higher reaches of set theory, although without physical applications, do carry ontological commitment by virtue of the fact that they have applications in other parts of mathematics. So long as the chain of applications eventually ``bottoms out`` in physical science, we could rightfully claim that the whole chain carries ontological commitment. Quine himself justifies some transfinite set theory along these lines (Quine 1984, p. 788), but he sees no reason to go beyond the constructible sets (Quine 1986, p. 400). His reasons for this restriction, however, have little to do with the indispensability argument and so supporters of this argument need not side with Quine on this issue.

3. Naturalism and Holism
Although both premises of the Quine-Putnam indispensability argument have been questioned, it`s the first premise that is most obviously in need of support. This support comes from the doctrines of naturalism and holism.
Following Quine, naturalism is usually taken to be the philosophical doctrine that there is no first philosophy and that the philosophical enterprise is continuous with the scientific enterprise (Quine 1981b). By this Quine means that philosophy is neither prior to nor privileged over science. What is more, science, thus construed (i.e. with philosophy as a continuous part) is taken to be the complete story of the world. This doctrine arises out of a deep respect for scientific methodology and an acknowledgment of the undeniable success of this methodology as a way of answering fundamental questions about all nature of things. As Quine suggests, its source lies in ``unregenerate realism, the robust state of mind of the natural scientist who has never felt any qualms beyond the negotiable uncertainties internal to science`` (Quine 1981b, p.72). For the metaphysician this means looking to our best scientific theories to determine what exists, or, perhaps more accurately, what we ought to believe to exist. In short, naturalism rules out unscientific ways of determining what exists. For example, naturalism rules out believing in the transmigration of souls for mystical reasons. Naturalism would not, however, rule out the transmigration of souls if our best scientific theories were to require the truth of this doctrine.[4]

Naturalism, then, gives us a reason for believing in the entities in our best scientific theories and no other entities. Depending on exactly how you conceive of naturalism, it may or may not tell you whether to believe in all the entities of your best scientific theories. I take it that naturalism does give us some reason to believe in all such entities, but that this is defeasible. This is where holism comes to the fore: in particular, confirmational holism.

Confirmational holism is the view that theories are confirmed or disconfirmed as wholes (Quine 1980b, p. 41). So, if a theory is confirmed by empirical findings, the whole theory is confirmed. In particular, whatever mathematics is made use of in the theory is also confirmed (Quine 1976, pp. 120-122). Furthermore, as Putnam (1979a) has stressed, it is the same evidence that is appealed to in justifying belief in the mathematical components of the theory that is appealed to in justifying the empirical portion of the theory (if indeed the empirical can be separated from the mathematical at all). Naturalism and holism taken together then justify P1. Roughly, naturalism gives us the ``only`` and holism gives us the ``all`` in P1.

It is worth noting that in Quine`s writings there are at least two holist themes. The first is the confirmational holism discussed above (often called the Quine-Duhem thesis). The other is semantic holism which is the view that the unit of meaning is not the single sentence, but systems of sentences (and in some extreme cases the whole of language). This latter holism is closely related to Quine`s well-known denial of the analytic-synthetic distinction (Quine 1980b) and his equally famous indeterminacy of translation thesis (Quine 1960). Although for Quine, semantic holism and confirmational holism are closely related, there is good reason to distinguish them, since the former is generally thought to be highly controversial while the latter is considered relatively uncontroversial.

Why this is important to the present debate is that Quine explicitly invokes the controversial semantic holism in support of the indispensability argument (Quine 1980b, pp. 45-46). Most commentators, however, are of the view that only confirmational holism is required to make the indispensability argument fly (see, for example, Colyvan (1998); Field (1989, pp. 14-20); Hellman (199?); Resnik (1995a; 1997); Maddy (1992)) and my presentation here follows that accepted wisdom. It should be kept in mind, however, that while the argument, thus construed, is Quinean in flavor it is not, strictly speaking, Quine`s argument.

4. Objections
There have been many objections to the indispensability argument, including Charles Parsons` (1980) concern that the obviousness of basic mathematical statements is left unaccounted for by the Quinean picture and Philip Kitcher`s (1984, pp. 104-105) worry that the indispensability argument doesn`t explain why mathematics is indispensable to science. The objections that have received the most attention, however, are those due to Hartry Field, Penelope Maddy and Elliott Sober. In particular, Field`s nominalisation program has dominated recent discussions of the ontology of mathematics.
Field (1980) presents a case for denying the second premise of the Quine-Putnam argument. That is, he suggests that despite appearances mathematics is not indispensable to science. There are two parts to Field`s project. The first is to argue that mathematical theories don`t have to be true to be useful in applications, they need merely to be conservative. (This is, roughly, that if a mathematical theory is added to a nominalist scientific theory, no nominalist consequences follow that wouldn`t follow from the nominalist scientific theory alone.) This explains why mathematics can be used in science but it does not explain why it is used. The latter is due to the fact that mathematics makes calculation and statement of various theories much simpler. Thus, for Field, the utility of mathematics is merely pragmatic - mathematics is not indispensable after all.

The second part of Field`s program is to demonstrate that our best scientific theories can be suitably nominalised. That is, he attempts to show that we could do without quantification over mathematical entities and that what we would be left with would be reasonably attractive theories. To this end he is content to nominalise a large fragment of Newtonian gravitational theory. Although this is a far cry from showing that all our current best scientific theories can be nominalised, it is certainly not trivial. The hope is that once one sees how the elimination of reference to mathematical entities can be achieved for a typical physical theory, it will seem plausible that the project could be completed for the rest of science.[5]

There has been a great deal of debate over the likelihood of the success of Field`s program but few have doubted its significance. Recently, however, Penelope Maddy, has pointed out that if P1 is false, Field`s project may turn out to be irrelevant to the realism/anti-realism debate in mathematics.

Maddy presents some serious objections to the first premise of the indispensability argument (Maddy 1992; 1995; 1997). In particular, she suggests that we ought not have ontological commitment to all the entities indispensable to our best scientific theories. Her objections draw attention to problems of reconciling naturalism with confirmational holism. In particular, she points out how a holistic view of scientific theories has problems explaining the legitimacy of certain aspects of scientific and mathematical practices. Practices which, presumably, ought to be legitimate given the high regard for scientific practice that naturalism recommends. It is important to appreciate that her objections, for the most part, are concerned with methodological consequences of accepting the Quinean doctrines of naturalism and holism - the doctrines used to support the first premise. The first premise is thus called into question by undermining its support.

Maddy`s first objection to the indispensability argument is that the actual attitudes of working scientists towards the components of well-confirmed theories vary from belief, through tolerance, to outright rejection (Maddy 1992, p. 280). The point is that naturalism counsels us to respect the methods of working scientists, and yet holism is apparently telling us that working scientists ought not have such differential support to the entities in their theories. Maddy suggests that we should side with naturalism and not holism here. Thus we should endorse the attitudes of working scientists who apparently do not believe in all the entities posited by our best theories. We should thus reject P1.

The next problem follows from the first. Once one rejects the picture of scientific theories as homogeneous units, the question arises whether the mathematical portions of theories fall within the true elements of the confirmed theories or within the idealized elements. Maddy suggests the latter. Her reason for this is that scientists themselves do not seem to take the indispensable application of a mathematical theory to be an indication of the truth of the mathematics in question. For example, the false assumption that water is infinitely deep is often invoked in the analysis of water waves, or the assumption that matter is continuous is commonly made in fluid dynamics (Maddy 1992, pp. 281-282). Such cases indicate that scientists will invoke whatever mathematics is required to get the job done, without regard to the truth of the mathematical theory in question (Maddy 1995, p. 255). Again it seems that confirmational holism is in conflict with actual scientific practice, and hence with naturalism. And again Maddy sides with naturalism. (See also Parsons (1983) for some related worries about Quinean holism.) The point here is that if naturalism counsels us to side with the attitudes of working scientists on such matters, then it seems that we ought not take the indispensability of some mathematical theory in a physical application as an indication of the truth of the mathematical theory. Furthermore, since we have no reason to believe that the mathematical theory in question is true, we have no reason to believe that the entities posited by the (mathematical) theory are real. So once again we ought to reject P1.

Maddy`s third objection is that it is hard to make sense of what working mathematicians are doing when they try to settle independent questions. These are questions, that are independent of the standard axioms of set theory - the ZFC axioms.[6] In order to settle some of these questions, new axiom candidates have been proposed to supplement ZFC, and arguments have been advanced in support of these candidates. The problem is that the arguments advanced seem to have nothing to do with applications in physical science: they are typically intra-mathematical arguments. According to indispensability theory, however, the new axioms should be assessed on how well they cohere with our current best scientific theories. That is, set theorists should be assessing the new axiom candidates with one eye on the latest developments in physics. Given that set theorists do not do this, confirmational holism again seems to be advocating a revision of standard mathematical practice, and this too, claims Maddy, is at odds with naturalism (Maddy 1992, pp. 286-289).

Although Maddy does not formulate this objection in a way that directly conflicts with P1 it certainly illustrates a tension between naturalism and confirmational holism.[7] And since both these are required to support P1, the objection indirectly casts doubt on P1. Maddy, however, endorses naturalism and so takes the objection to demonstrate that confirmational holism is false. I`ll leave the discussion of the impact the rejection of confirmational holism would have on the indispensability argument until after I outline Sober`s objection, because Sober arrives at much the same conclusion.

Elliott Sober`s objection is closely related to Maddy`s second and third objections. Sober (1993) takes issue with the claim that mathematical theories share the empirical support accrued by our best scientific theories. In essence, he argues that mathematical theories are not being tested in the same way as the clearly empirical theories of science. He points out that hypotheses are confirmed relative to competing hypotheses. Thus if mathematics is confirmed along with our best empirical hypotheses (as indispensability theory claims), there must be mathematics-free competitors. But Sober points out that all scientific theories employ a common mathematical core. Thus, since there are no competing hypotheses, it is a mistake to think that mathematics receives confirmational support from empirical evidence in the way other scientific hypotheses do.

This in itself does not constitute an objection to P1 of the indispensability argument, as Sober is quick to point out (Sober 1993, p. 53), although it does constitute an objection to Quine`s overall view that mathematics is part of empirical science. As with Maddy`s third objection, it gives us some cause to reject confirmational holism. The impact of these objections on P1 depends on how crucial you think confirmational holism is to that premise. Certainly much of the intuitive appeal of P1 is eroded if confirmational holism is rejected. In any case, to subscribe to the conclusion of the indispensability argument in the face of Sober`s or Maddy`s objections is to hold the position that it`s permissible at least to have ontological commitment to entities that receive no empirical support. This, if not outright untenable, is certainly not in the spirit of the original Quine-Putnam argument.

5. Conclusion
It is not clear how damaging the above criticisms are to the indispensability argument. Indeed, the debate is very much alive, with many recent articles devoted to the topic. (See bibliography notes below.) Closely related to this debate is the question of whether there are any other decent arguments for platonism. If, as some believe, the indispensability argument is the only argument for platonism worthy of consideration, then if it fails, platonism in the philosophy of mathematics seems bankrupt. Of relevance then is the status of other arguments for and against mathematical realism. In any case, it is worth noting that the indispensability argument is one of a small number of arguments that have dominated discussions of the ontology of mathematics. It is therefore important that this argument not be viewed in isolation.
The two most important arguments against mathematical realism are the epistemological problem for platonism - how do we come by knowledge of causally inert mathematical entities? (Benacerraf 1983b) - and the indeterminacy problem for the reduction of numbers to sets - if numbers are sets, which sets are they (Benacerraf 1983a)? Apart from the indispensability argument, the other major argument for mathematical realism is that it is desirable to provide a uniform semantics for all discourse: mathematical and non-mathematical alike (Benacerraf 1983b). Mathematical realism, of course, meets this challenge easily, since it explains the truth of mathematical statements in exactly the same way as in other domains.[8] It is not so clear, however, how nominalism can provide a uniform semantics.

Finally, it is worth stressing that even if the indispensability argument is the only good argument for platonism, the failure of this argument does not necessarily authorize nominalism, for the latter too may be without support. It does seem fair to say, however, that if the objections to the indispensability argument are sustained then one of the most important arguments for platonism is undermined. This would leave platonism on rather shaky ground.[9]

Bibliography
Although the indispensability argument is to be found in many places in Quine`s writings (including 1976; 1980a; 1980b; 1981a; 1981c), the locus classicus is Putnam`s short monograph Philosophy of Logic (included as a chapter of the second edition of the third volume of his collected papers (Putnam, 1979b)). See also Putnam (1979a) and the introduction of Field (1989) which has an excellent outline of the argument. Colyvan (2001) is a sustained defence of the argument.
See Chihara (1973), and Field (1980; 1989) for attacks on the second premise and Colyvan (1999b; 2001), Maddy (1990), Malament (1982), Resnik (1985), Shapiro (1983) and Urquhart (1990) for criticisms of Field`s program. For a fairly comprehensive look at nominalist strategies in the philosophy of mathematics (including a good discussion of Field`s program), see Burgess and Rosen (1997), while Feferman (1993) questions the amount of mathematics required for empirical science. See Azzouni (1997), Balaguer (1996b; 1998), Leng (2002), Maddy (1992; 1995; 1997), Melia (2000), Peressini (1997), Sober (1993) and Vineberg (1996) for attacks on the first premise. Colyvan (1998; 1999a; 2001; 2002), Hellman (1999) and Resnik (1995a; 1997) reply to some of these objections.

For variants of the Quinean indispensability argument see Maddy (1992) and Resnik (1995a).

Azzouni, J., 1997, ``Applied Mathematics, Existential Commitment and the Quine-Putnam Indispensability Thesis``, Philosophia Mathematica (3) 5/3 (October): 193-209
Balaguer, M., 1996a, ``Towards a Nominalization of Quantum Mechanics``, Mind 105/418 (April): 209-226
Balaguer, M., 1996b, ``A Fictionalist Account of the Indispensable Applications of Mathematics``, Philosophical Studies 83/3 (September): 291-314
Balaguer, M., 1998, Platonism and Anti-Platonism in Mathematics, New York: Oxford University Press
Benacerraf, P., 1983a, ``What Numbers Could Not Be``, reprinted in Benacerraf and Putnam (1983), pp. 272-294
Benacerraf, P., 1983b, ``Mathematical Truth``, reprinted in Benacerraf and Putnam (1983), pp. 403-420 and in Hart (1996), pp. 14-30
Benacerraf, P. and Putnam, H. (eds.), 1983, Philosophy of Mathematics: Selected Readings, 2nd edition, Cambridge: Cambridge University Press
Burgess, J., 1983, ``Why I Am Not a Nominalist``, Notre Dame Journal of Formal Logic 24/1 (January): 93-105
Burgess, J. and Rosen, G., 1997, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics, Oxford: Clarendon
Chihara, C., 1973, Ontology and the Vicious Circle Principle, Ithaca, NY: Cornell University Press
Colyvan, M., 1998, ``In Defence of Indispensability``, Philosophia Mathematica (3) 6/1 (February): 39-62
Colyvan, M., 1999a, ``Contrastive Empiricism and Indispensability``, Erkenntnis 51/2-3 (September): 323-332
Colyvan, M., 1999b, ``Confirmation Theory and Indispensability``, Philosophical Studies 96/1 (October): 1-19
Colyvan, M., 2001, The Indispensability of Mathematics, New York: Oxford University Press
Colyvan, M., 2002, ``Mathematics and Aesthetic Considerations in Science``, Mind 111/441 (January): 69-74
Feferman, S., 1993, ``Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics``, Proceedings of the Philosophy of Science Association 2: 442-455
Field, H.H., 1980, Science Without Numbers: A Defence of Nominalism, Oxford: Blackwell
Field, H.H., 1989, Realism, Mathematics and Modality, Oxford: Blackwell
Hart, W.D. (ed.), 1996, The Philosophy of Mathematics, Oxford: Oxford University Press
Hellman, G., 1999, ``Some Ins and Outs of Indispensability: A Modal-Structural Perspective``, in A. Cantini, E. Casari and P. Minari (eds.), Logic and Foundations of Mathematics, Dordrecht: Kluwer, pp. 25-39
Irvine, A.D. (ed.), 1990, Physicalism in Mathematics, Dordrecht: Kluwer
Kitcher, P., 1984, The Nature of Mathematical Knowledge, New York: Oxford University Press
Leng, M., 2002, ``What`s Wrong with Indispensability? (Or, The Case for Recreational Mathematics)``, Synthese 131/3 (June): 395-417
Maddy, P., 1990, ``Physicalistic Platonism``, in A.D. Irvine (ed.), Physicalism in Mathematics, Dordrecht: Kluwer, pp. 259-289
Maddy, P., 1992, ``Indispensability and Practice``, Journal of Philosophy 89/6 (June): 275-289
Maddy, P., 1995, ``Naturalism and Ontology``, Philosophia Mathematica (3) 3/3 (September): 248-270
Maddy, P., 1997, Naturalism in Mathematics, Oxford: Clarendon Press
Maddy, P., 1998, ``How to be a Naturalist about Mathematics``, in H.G. Dales and G. Oliveri (eds.), Truth in Mathematics, Oxford: Clarendon, pp. 161-180
Malament, D., 1982, ``Review of Field`s Science Without Numbers``, Journal of Philosophy 79/9 (September): 523-534 and reprinted in Resnik (1995b), pp. 75-86
Melia, J., 2000, ``Weaseling Away the Indispensability Argument``, Mind 109/435 (July): 455-479
Parsons, C., 1980, ``Mathematical Intuition``, Proceedings of the Aristotelian Society 80 (1979-1980): 145-168 and reprinted in Resnik (1995b), pp. 589-612 and in Hart (1996), pp. 95-113
Parsons, C., 1983, ``Quine on the Philosophy of Mathematics``, in Mathematics in Philosophy: Selected Essays, Ithaca, NY: Cornell University Press, pp. 176-205
Peressini, A., 1997, ``Troubles with Indispensability: Applying Pure Mathematics in Physical Theory``, Philosophia Mathematica (3) 5/3 (October): 210-227
Putnam, H., 1979a, ``What is Mathematical Truth``, in Mathematics Matter and Method: Philosophical Papers Vol. 1, 2nd edition, Cambridge: Cambridge University Press, pp. 60-78
Putnam, H., 1979b, ``Philosophy of Logic``, reprinted in Mathematics Matter and Method: Philosophical Papers Vol. 1, 2nd edition, Cambridge: Cambridge University Press, pp. 323-357
Quine, W.V., 1960, Word and Object, Cambridge, MA: Massachusetts Institute of Technology Press
Quine, W.V., 1976, ``Carnap and Logical Truth`` reprinted in The Ways of Paradox and Other Essays, revised edition, Cambridge, MA: Harvard University Press, pp. 107-132 and in Benacerraf and Putnam (1983), pp. 355-376
Quine, W.V., 1980a, ``On What There Is``, reprinted in From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard University Press, pp. 1-19
Quine, W.V., 1980b, ``Two Dogmas of Empiricism``, reprinted in From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard University Press, pp. 20-46 and in Hart (1996), pp. 31-51 (Page references are to the first reprinting)
Quine, W.V., 1981a, ``Things and Their Place in Theories``, in Theories and Things, Cambridge, MA: Harvard University Press, pp. 1-23
Quine, W.V., 1981b, ``Five Milestones of Empiricism``, in Theories and Things, Cambridge, MA: Harvard University Press, pp. 67-72
Quine, W.V., 1981c, ``Success and Limits of Mathematization``, in Theories and Things, Cambridge, MA: Harvard University Press, pp. 148-155
Quine, W.V., 1984, ``Review of Parsons`, Mathematics in Philosophy``, Journal of Philosophy 81/12 (December): 783-794
Quine, W.V., 1986, ``Reply to Charles Parsons``, in L. Hahn and P. Schilpp (eds.), The Philosophy of W.V. Quine, La Salle, ILL: Open Court, pp. 396-403
Resnik, M.D., 1985, ``How Nominalist is Hartry Field`s Nominalism``, Philosophical Studies 47 (March): 163-181
Resnik, M.D., 1995a, ``Scientific Vs Mathematical Realism: The Indispensability Argument``, Philosophia Mathematica (3) 3/2 (May): 166-174
Resnik, M.D. (ed.), 1995b, Mathematical Objects and Mathematical Knowledge, Aldershot (UK): Dartmouth
Resnik, M.D., 1997, Mathematics as a Science of Patterns, Oxford: Clarendon Press
Shapiro, S., 1983, ``Conservativeness and Incompleteness``, Journal of Philosophy 80/9 (September): 521-531 and reprinted in Resnik (1995b), pp. 87-97 and in Hart (1996), pp. 225-234
Sober, E., 1993, ``Mathematics and Indispensability``, Philosophical Review 102/1 (January): 35-57
Urquhart, A., 1990, ``The Logic of Physical Theory``, in A.D. Irvine (ed.), Physicalism in Mathematics, Dordrecht: Kluwer, pp. 145-154
Vineberg, S., 1996, ``Confirmation and the Indispensability of Mathematics to Science`` PSA 1996 (Philosophy of Science, supplement to vol. 63), pp. 256-263

#133 by DRUMZ on January 24, 2004 3:39pm PT

http://users.forthnet.gr/ath/kimon/philosophy.htm

Do Mathematics Exist?

There are basically two mainstream schools of thought, (Neo-) Platonism and Formalism, and a third somewhat heretical view, Constructivism.

According to Neo-Platonists, mathematics exist independent of human quest, so they are in fact discovered, not invented. Even the most abstract mathematical objects are real and invariable, immaterial of course and in no way related to physical existence, space and time, but anyway they do exist in a non-objective world. For a Neo-Platonist, there is an answer to Cantor`s Continuum Hypothesis, only we do not have the means to obtain it, that is, we do not understand real numbers sufficiently.

A strong argument in favor of the Neo-Platonic view is the ``unreasonable effectiveness of mathematics in the natural sciences`` (Eugene Wigner, Nobel prize-winning physicist). Another one is given by the Russian mathematician I. Shafarevitch:

``History of mathematics has known many occasions where a discovery made by a scientist remains unknown until somebody else makes it again later, with astonishing preciseness. In the letter that Galois wrote the day before his fatal duel, he reached some conclusions of extreme importance in the study of integrals of algebraic functions. More than twenty years later, Riemann, undoubtedly unaware of Galois` letter, re-discovered and proved the same propositions. Another example: after Lobachevski and Bolyai built the foundations of non-Euclidean geometry independent of each other, it appeared that two other mathematicians, Gauss and Schweikart, had both reached the same conclusions ten years earlier, also working independently. There is a strange feeling in reading exactly the same ideas, as coming from one mind, in the work of four scientists who studied the subject alone`` (talk given to the Göttingen Academy of Sciences, 1973)


For Formalists on the other, mathematical objects do not exist. Mathematics consist of symbols, axioms/sentences composed of such symbols and rules to transform sentences into others (e.g. theorems), but none of these has any particular meaning. Mathematics is therefore a humanly constructed language devised by human beings for definite ends prescribed by themselves.

Formalists often speak in terms of Neo-Platonic real objects, but only for reasons of convenience. For a Formalist, the Cantor`s Continuum Hypothesis is meaningless for there is no such a thing as a complete understanding of the real numbers. As long as we follow the strict rules of transforming series of symbols into other series of symbols, there is no point in asking whether we approach reality or not, because there is no reality.


Constructivists take the extreme view that if something cannot be constructed in a finite number of steps it does not exist. The leader was L.E.J. Brouwer who even devised a famous counter-example to show that the trichotomy law for real numbers (every real number is either negative, zero or positive) is not true. The argument involved a strictly defined but impossible calculation with Pi the result of which would define in turn the sign of a related number. Constructivists would dispose all questions about infinity on these grounds.

Links on the Logic and Philosophy of Mathematics can be found at the University of Waterloo/ Department of Philosophy site here

What Does ``Existence`` Mean in Mathematics?
Understanding what existence means in mathematics is the key to understanding what it means for concepts like ``infinity`` or ``imaginary numbers`` to exist--something that puzzles a lot of people when they first encounter these weird ideas!
Mathematical objects do not exist in the same sense that a physical object exists; nobody has ever bumped their elbow on a number, for instance.

Instead, mathematical objects are abstract concepts (often abstracted from a real world situation, by isolating just the part of the situation that is relevant for a particular discussion).

When we ask whether or not a mathematical object exists, we must have in mind an appropriate context: a particular, precisely defined collection of concepts. Then we ask, ``among these concepts, is there one which matches the object we are looking for?`` If so, we say that the object exists; if not, it doesn`t exist.

For example, the natural numbers (that is, the numbers 1, 2, 3, 4, and so on) are the concepts obtained by abstracting the property of ``size`` from collections of objects. The number 2 is the abstract concept that expresses what the following collections of objects have in common: the eyes on a person`s face, the occurrences of the letter ``b`` in the word ``bib``, the wheels on a bicycle, and so on.

If we were to ask ``does there exist a natural number between 1 and 2``, we mean, ``among the collection of natural numbers, is there one (say x) such that 1 < x < 2?`` The answer to this question is ``No``. You cannot, for example, go to the beach and pick up more than one pebble but fewer than two pebbles.

If we were to ask ``does there exist a number between 1 and 2``, the answer would depend on what we meant by the word number.

If we were using the word ``number`` to mean ``natural number`` (that is, measurement of the size of a set), then the answer would be ``No; no such number exists.``

However, there are other contexts in which the answer might be ``Yes``. For example, we might be in a context where number is meant to refer, not to a natural number, but to a rational number: that is, a fraction.

Rational numbers are something quite different from natural numbers: instead of being measurements of sizes of sets, they are ratios of the sizes of two sets. For example, the fraction 3/4 is expressing the ratio ``3 to 4``.

In this context, where number refers to a ratio not to the size of a single set, and where ``2`` and ``1`` really mean the fractions ``2/1`` and ``3/1`` respectively, then the answer to our question is ``yes``: there does exist a number between 1 and 2, for instance the fraction 3/2.

This illustrates that, before one can say whether a concept exists or not, we have to be quite clear about the context in which the question is being asked.

If you are still puzzled by this, you might want to read more (in a discussion of whether or not ``imaginary numbers`` exist) about how there are many quite different meanings for the word ``number``, and how whether or not a concept exists can depend on the meaning you have in mind.