A discussion on Ontology of numbers

Summary of Collins A.: ON THE QUESTION ‘DO NUMBERS EXIST?’ in The Philosophical Quarterly, Vol. 48, No. 190 (1998). pp. 23-36

1.0 Introduction

In human thought, the numbers appear to have originated from the various physical situations encountered by man – the difference between a herd and a goat, the difference between 3 deers and 1 deer, the correspondence between number of shadows and number of people etc. At some point, the abstract property which is common to say two groups (ex. 5 arrows and 5 birds) was recognized and this is what is called a number. This is a primitive notion of a number, in contemporary times we have various kinds of numbers – the rationals, the irrationals, the imaginary numbers etc. But all the kinds of numbers are equally abstract. So the question arises, where does the number exist or do they exist at all. The paper is an attempt to this question. The author treats the problem as of having a linguistic origin. The author rejects the usual answers to the problem that is realism and nominalism, so first I will explain these usual answers in brief before presenting authors views.

2.0 Realism and Nominalism

“Realism is a philosophy of mathematics and an ontological commitment” (Collins, pp23). That is, realism is the belief that properties, usually called Universals, exist independently of the things that manifest them. Therefore, if we remove all the rectangular shaped objects from the universe, still the universal rectangle will exist. Thus it can be seen that there are two aspects to realism. First, there is a claim about existence. The billiard tables, the football grounds, the books exist and so does the rectangleness. The second aspect of realism concerns independence. The fact that the billiard table exists and is rectangular is independent of anything anyone happens to say or think about the matter. The question of the nature and plausibility of realism is a controversial issue (Miller, 2002).

Nominalism is an anti-realist stand. The doctrine holding that abstract concepts, general terms, or universals have no independent existence but exist only as names. The relation between universal and name is conventional.

But there is a problem with nominalism as pointed out by Prof. Gomatam during the class discussion; it is the fact that a universal say rectangleness cannot be attributed to arbitary things, so universals are not just names.

3.0 The short argument

It is a familiar thought that we might posit numbers to explain the known arithmetical truths and scientific truths the expression of which requires numerical representation (Collins, pp. 23). To explain this author gives an example,

There are four prime numbers smaller that 8.

If this fact is known, then we know the numbers exist. This is what author calls the short argument for the existence of numbers. Further, the author believes that if there were no debate between realism and nominalism the short argument would be satisfactory. I am unable to agree with this point. First of all, the condition of ‘primality’ comes into discussion when we talk about numbers. It seems that first we have numbers and then we define the condition of primality on them. Thus, the fact that “there are four prime numbers less that 8”, cannot be known without the prior knowledge of numbers. It seems that the short argument is trying to prove the existence of numbers in the hindsight i.e. by at first assuming them. I think Quine is also trying to point to this fact in the following quote, “…indispensability of mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes”(Quine as quoted by Putnam in Collins,1998, pp. 26). I think it is right to say that if numbers exist it means that ‘there are four prime numbers smaller than 8’. Although, this does not answer the original question but it shows the problem in the short argument. Even if we consider the scientific truths, for example, the velocity of light is 3 X 10⁸, the same problem arises. The short argument says that since we know this fact, it implies the numbers exist. But the concept of velocity (=distance/time) comes into discussion only if we have prior knowledge of numbers. Author claims that the short argument leaves no space for positing of the numbers. I he knows that there are four primes less than 8 it entails that he already has the knowledge of numbers and he need not posit them, whereas, both realism and nominalism posit numbers.

Quine is a soft realist. As quoted by Putnam, Quine thinks it is intellectual dishonesty to use and talk about mathematical entities, yet deny their existence. Quine treats everything as a myth, but he treats the myth of physical objects as superior to say Homer’s gods. He says, “…The myth of physical objects is epistemologically superior to most in that it has proved more efficacious as a device for working a manageable structure into the flux of experience”. Hartry Field also has a similar view. If believes that if numbers can be shown to be a useful myth then it can be shown that they are fictional thus, they do not exist.

4.0 Linguistic solution

Author says that short argument is sufficient to show the existence of numbers but the question about the way they exist, should not be asked as it is inappropriate. He accepts that numbers do not exist as the physical objects do, still it is right to say that they exist. As pointed by Prof. Gomatam, it seems what author wants to say is that by asking the question about way the numbers exist, we are making a category mistake. For example, if pen is in the pocket, we can ask someone to pull it out. Whereas, if someone says, he has an idea in his head we do not ask him to pull it out.

Another problem while thinking about the existence of numbers is that they are defined by giving them negative attributes like they are non-physical, non-spatial, non-temporal, non-corrutible, non-contingent and they do not enter the causal relationships(Collins, pp. 30).

5.0 Conclusion

Although, I am unable to agree that the short argument is sufficient to establish the existence of numbers. But I am inclined to believe that the numbers exist based on Quine’s argument. Field also recognizes that Quine’s argument is the strongest argument for realism. Quine argues that it is indispensable to talk about mathematical entities yet their existence is denied which implies that there is an intellectual dishonesty. Yet, the question of about what is for them to exist is not answered. Author argues that the question is wrong because our discourse with numbers does not generate the question. Also, because of the fact that the numbers are given negative attributes, the question of what is for them to exist sounds wrong to me.

6.0 References:

1. Miller, A (2002), Stanford Encyclopedia of Philosophy, http://setis.library.usyd.edu.au/stanford/entries/realism/.

2. Collins, A.(1998, ON THE QUESTION ‘DO NUMBERS EXIST?’ in The Philosophical Quarterly, Vol. 48, No. 190 pp. 23-36.

SCIENTIFIC REALISM AND QUANTUM PHYSICS

Summary of Priest G., “Primary Qualities Are Secondary Qualities Too”, British Journal for Philosophy of Science, Vol. 40, (1989), pp. 29-37.

1.0 Introduction

In the paper, the author is comparing the current conflict between quantum physics and scientific realism with the scientific revolution of the 17^th century. According to the author, quantum physics is indicating a change in the 17^th century scientific conception of matter. By doing so he is arguing that such a change in the conception of matter will lead to a realistic interpretation of quantum mechanics.

2.0 Primary and Secondary Qualities

The mechanistic conception of matter which was formed by the work of primarily Galileo and Descartes, characterizes matter by its extension and locatability in space and time. These are what are called primary properties. Matter would have these properties even if there is no conscious observer present. But some properties of matter like color, smell etc. will not be present without the presence of conscious observer. Such properties are called secondary properties and they arise because of the interaction between an observer and the object.

With the advent of atomic and wave theories, it was possible to show that the dispositions which lead to the rise of secondary properties where really aggregate primary properties of micro-structure of matter.

According to the author, similar kind of revision in the conception of matter is indicated by quantum mechanics. In quantum mechanics, some properties like the coefficients of the eigenstates are observer-independent and hence such prperties are analougous to the primary properties of the mechanistic conception of matter. Whereas, some properties like spin which are primary in the mechanistic conception are observer-dependent and such properties are analougous to the secondary properties of the mechanistic conception.

3.0 EPR

EPR argument brings out the strongest objection to realism. According to realism, the happenings at one place cannot affect the happenings at other places instantaneously, whereas, EPR seems to say the opposite.

The author argues that the problem arises when the idea that there are two particles (in context of EPR) interacting in space-time is forced on to the situation. In fact, there is no problem for realism if we accept that Ψ state is what describes reality and thus there are no to particles out there.

How Physicalism And ‘Common Sense’ Description Of The World Can Be Made Compatible?

1.0 Introduction

It is commonly known that physical theories conflict with our ordinary common sense views. For example, it is our ordinary experience that sun revolves around the earth; whereas, the scientific theory says that both sun and earth revolve about a point called center of mass (for the case of sun and earth, this approximately means that earth revolves around the sun, which is opposite to our ordinary experience). In the paper, the author is arguing for an interpretation of physicalism which is compatible with common sense.

The author argues that scientific discoveries cannot contradict in any fundamental way the tenets of common sense that are based on ordinary experience, if we believe that the scientific investigation involves a refinement of common sense. And, the scientific discoveries can only undermine those of our ordinary views about the world that are based on inadequate or distorted observation. I am unable to agree with this argument. It is our ordinary experience that we have free will (i.e. we have capability to make choices and decide among of them); I think this experience is neither inadequate nor distorted observation. But no physical theory can account for it; rather current physical theories reject it as entirely deceptive.

2.0 Tentative Realism

Physics is a precise discipline, that is, at any time most physicists agree as to which theories are acceptable. But there is no general agreement on the kind of interpretation to be given to the mathematical formalism of a physical theory.
















Now the question arises that does a successful mathematical formalism given a physicalist interpretation, constitute a possible theory of physics, as opposed perhaps to a theory of metaphysics. To this the author answers by presenting Popper’s solution of demarcation between physics and meta-physics on the basis of experimental falsifiability.

That is, the theory belongs to physics if it is experimentally falsifiable. From this requirement it become clear that the kind of interpretation required for the mathematical formalism demanded by physicalism is what is called ‘tentative realism’. It states that the fact that the theory must be open to experimental refutation ensures that it is meaningful to call a theory false, which in turn ensures that it must be meaningful to call the theory true.

3.0 An acceptable Physicalism

Author’s main aim in the paper is to present a kind of physicalism that is compatible with common sense. In order to do that, I think he is using the concept of drawing distinctions propounded by Spencer-Brown in his book ‘Laws of form’. Physical theory and common sense theory draw different kinds of distinctions in the world. Thus, they classify things in terms of different kinds of resemblances between things. Author suggests that the Physicalism that is needed in the present context should classify things in the following way:

1. The things are classified in the simplest possible way i.e. in terms of causal sequences.
2. The things are classified in terms of only those resemblances which any intelligent being, however its sensory equipment may be constructed can discern, discover, become aware of. That is, classification is sense-independent.

On the other hand, common sense theory classifies things in terms of resemblances which are discernible to human beings or are associated with their experiences.

Now, common sense theory has a property called ‘color’. This property is discernible to humans because of their sensory equipment. But it falls out of the periphery of physicalism if it satisfies the above two requirements. So in this way the common sense theory and physicalist theory is compatible.

References:

1. http://www.nick-maxwell.demon.co.uk/About%20Me.htm
2. Maxwell N. (1965: May – 1966: Feb), “PHYSICS AND COMMON SENSE”, British Journal for Philosophy of Science, Vol. 16, pp. 295-311.

Why Einstein made the statement that “God does not Play dice”?

1.0 Introduction

Apart from the fact that Einstein contributed fundamentally to physics, he is also known for his life-long opposition to the most successful physical theory – the quantum theory. In this context, he made his famous statement “God does not play dice [with the universe]”. It is difficult to find the exact reasons why the man who at first said that quantum theory was revolutionary, later found objections to the developments in field (especially, Copenhagen interpretation). The author tries to uncover the reasons which motivated Einstein to take this stand. In order to do that the author presents Einstein’s conception of scientific realism (which is also the title of the paper).

2.0 Einstein a Realist

We will start with a quote by Einstein, “It is basic for physics that one assumes a real world existing independently from any act of perception” (Einstein, quoted in Gomatam, pp. 4). From this quote it appears that Einstein was a realist in the sense it is commonly accepted. But on further analysis we find that such a conclusion is not tenable. As Einstein said, “I agree physics concerns the ‘real’, but I am not a realist.

Einstein differs from the conventional views of scientific realism on two counts. First, is that he was not interested in the “context of justification”. He was more interested in the in the relation between physics and reality at the level of theory creation i.e. “context of creation”. Secondly, even at the level of theory creation his conception of realism involves, not the relation between theory and reality, but between the physicist and the reality (Gomatam, pp. 1).

Einstein’s claim is that physics is the attempt at the conceptual construction of a model of the real world and of its lawful structure and that by means of conceptual thinking we can grasp reality. Now, we can see some insights into a prospective solution to the original question of why Einstein made the statement that “God does not Play dice”? According to Einstein, reality can be grasped by means of conceptual thinking, but this is not possible in the Copenhagen interpretation of quantum physics.

3.0 Relation between creator-physicist and reality

The author brings out a very important point about the process of theory creation as viewed by physicists themselves. “Going by their [physicists] testimonies, the creation of a successful theory is grounded in a profound grasp of the physical reality that is neither purely conceptual nor deduced logically from experiences. It seems to be, as it were, a mystical insight into nature, ‘akin to religious worshipper’ as Plank puts it” (Gomatam, pp. 6).

According to Einstein, the physics concerns the real in the sense that physics is an effort of the physicist to express the grasp of reality that he has in his thinking through mathematically constructed concepts that have empirical usefulness. Author points out that Einstein’s realism concerns the relationship between the creator-physicist and reality. From this it is quite clear that there is component of subjectivity in the Einstein’s realism. MacKinnon remarks on this, “The unparalleled success of Einstein’s early efforts gave his realism an extremely personal quality…” Given this form of realism, it becomes clear why Einstein could not fully accept the Copenhagen interpretation because it settled for a probabilistic, non-visualizable account of physical behavior thus even the physicist(the creator of the theory) did not grasp the reality. Hence he said that, “Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’.”

4.0 Conclusion

The paper very clearly brings out some aspects of Einstein’s worldview which help in clarifying the reasons of his objection to quantum theory as the final theory. Although the fact that Einstein’s realism has a subjective component can be attacked. Plank and possibly Newton also had similar views. It is not possible to ignore their views as these physicists are in the highest realm among the physicists.

5.0 References:

1. Gomatam R. (2005) ‘Einstein’s Conception of Scientific Realism’, Unpublished Manuscript.

2. http://en.wikipedia.org/wiki/Einstein.

Worrall J.: ‘Structural Realism: The Best of Both Worlds?’

RECOVERING A WEAKER FORM OF REALISM – STRUCTURAL REALISM

1.0 INTRODUCTION

Scientific realism is a doctrine that the entities postulated by scientific theories are real entities in the world, with approximately the properties attributed to them by the best available scientific theories. The doctrine was widely accepted till the late nineteenth century primarily due to success of Newtonian mechanics. But with the developments in physical theories especially quantum physics and relativity, it became very difficult to hold on to the realist position. In the paper, author’s aim is to recover some sort of realism. He argues in favor of structural realism, a position held by Poincaré also. I myself tend to believe in realism. I aim to work in physics and I find it very difficult to accept that my work is not about the real world. I think it is appropriate to quote Plank here, “Why should anybody go to the trouble of gazing at the stars if he did not believe that the stars were really there? … .”(Plank, quoted in Gomatam, pp.5)

2.0 NO-MIRACLES ARGUMENT

This is the main argument for a realist. The argument says that it would be a miracle, if a theory made many correct empirical predictions without what that the theory says about the fundamental structure of the universe being correct or “essentially” correct. Since, miracles are not to be accepted, thus, it is plausible to conclude that presently accepted theories are “essentially” correct. The argument requires empirical predictions for which the theory has not been engineered. For example, the predictions about the bending of light rays when they pass near the sun, made by GTR.

Although the argument is forceful it runs into problems right away. Newton’s theory of gravitation had a wide range of predictive success – the return of Halley’s Comet, discovery of Neptune etc. So according to no-miracles argument, Newton’s theory shows the fundamental structure of the universe. But we know that Newton’s theory was rejected in favor of Einstein’s GTR. What is even more striking is that the two theories are logically inconsistent, that is, if GTR is true then Newton’s theory is false and vice versa. That is, Einstein’s theory is not a mere extension of Newton’s theory. This shows that the development of science is not cumulative. Thus, no-miracles argument is not sufficient to establish realism.

Here, the author points out that no present-day realist would claim that we have grounds for holding that presently accepted theories are true. Instead, a scaled downed version called modified realism is propounded. Modified realism says our present theories in mature science are approximately true.


3.0 PESSIMISTIC INDUCTION

This is the main argument against realism. The history of science is full of theories which at different times and for long periods had been empirically successful, and yet were shown to be false in the certain they made about the world. It is similarly full of theoretical terms featuring in successful theories which do not refer. Therefore, by a simple induction on scientific theories, our current successful theories are likely to be false, and many or most of the theoretical terms featuring in them will turn out to be non-referential. Therefore, the empirical success of a theory provides no warrant for the claim that the theory is approximately true. This argument is known as the Pessimistic Induction. It was first propounded by Poincaré.

If the above argument is accepted then undoubtedly realism is untenable. Here author shows two ways that can help to show that the picture of theory-change is wrong. First, the successful empirical content of the previously accepted theory is in general carried over to the new theory, but its basic theoretical claims are not. Thus, theories are best construed as making no claims beyond their empirical consequences. This position is called pragmatic anti-realism. The other alternative uses the fact that although scientific theories are not explanatorily cumulative but predicatively they are cumulative. Here, it is believed that the observation-transcendental parts of scientific theories are not just codification schemes, they are accepted descriptions of reality hidden behind the phenomena, and our present best theories are our presently best shots at truth. Yet, there is no reason to believe that those present theories are closer to truth than their rejected predecessors. This is known as conjectural realism given by Popper. As a realist, I find this position to be fairly acceptable. But scientists generally agree to the point that present theories are closer to the truth than previous ones. Another problem with this position is that conjectural realism makes no concessions to no miracles argument (the main argument for realism).


4.0 STRUCTURAL REALISM

So the aim now is to accommodate intuitions lying under no-miracles argument and historical facts about the theory change in science. Then the position will be more plausible than pragmatic anti-realism and conjectural realism. Structural Realism is such a position. To explain this position, I will give the example used by author. Fresnel’s theory of light was based on the assumption that light consists in periodic disturbances originating in a source and transmitted by an all-pervading mechanical medium (ether). Fresnel’s theory had predictive success like prediction of white spot at the center of the shadow of an opaque disc held in light. Thus, it will be accepted as a mature science. But after Maxwell’s theory, light became to be viewed as a periodic disturbance, not in an elastic medium, but in electromagnetic field. Yet there is continuity in the shift from Fresnel to Maxwell, but the continuity is of structure and not of content. Now, we can say that Fresnel completely misidentified the nature of light, but it is no miracle that this theory enjoyed predictive success, because his theory attributed the right structure to the light.


5.0 CONCLUSION

From the paper, I conclude that the doctrine of Structural Realism is able to accommodate the no-miracles argument and the pessimistic induction argument. This is important because both these arguments are very strong. Falsifying any of them is not easy. Thus, structural realism seems to me to be the most appropriate stand for a realist.


6.0 REFERENCES

1. Worrall J. (1989), ‘Structural Realism: The Best of Both Worlds?’, Dialectica, Vol. 43, pp.99-124.
2. Gomatam R. (2005), ‘Einstein’s Conception of Scientific Realism’, Unpublished manuscript.

Chaos Theory

A. BRIEF HISTORY

1. Father of chaos Theory – Henri Poincaré

Poincaré is regarded as the last "universalist" capable of understanding and contributing in virtually all parts of mathematics. He is considered to be the father of chaos theory, although the term ‘chaos’ was not coined by him. He was working to the famous three-body problem. Due to the efforts of Poincaré, today we know that the problem is not solvable analytically. But in 1889, this was not known. The three-body problem is a specific case of n-body problem (n>=3). A system consisting of two bodies which are interacting under gravitation forces can be expressed as a differential equations. Using Newton’s laws, the motions of two masses can be expressed as a differential equation. In mathematical parlance, it is said that the system is “analytically solvable” or the set of differential equation don’t have a “closed form solution”. Intuitively this indicates that long range prediction of orbits is not possible. Poincaré showed that a system of three bodies interacting under gravitational forces cannot be solved analytically. The solutions in such cases can be approximated by using computational methods.

2. Edward Lorentz

Poincaré’s monumental discovery of what is now called deterministic chaos was neglected. Probably, the time at which it came, scientists were primarily interested in Relativity theory and Quantum Physics. Also, without the computer it is very difficult to show the behavior of chaotic deterministic system.

If we are allowed to say that Poincaré gave birth to chaos theory, then it can be safely said that Lorentz gave it a rebirth. Edward Lorentz was a meteorologist working at MIT on long range prediction of weather using computer models. The models consisted of differential equations, no more complicated than Newton’s laws of motion. He created a basic computer program using mathematical equations which could theoretically predict what the weather might be. One day he wanted to run a particular sequence again, and to save time he started it from the middle of the sequence. After letting the sequence run he returned to find that the sequence had evolved completely different from the original. At first he couldn't comprehend such different results but then realized that he had started the sequence with his recorded results to 3 decimal places, whereas the computer had recorded them to 6 decimal places. As this program was theoretically deterministic we would expect a sequence very close to the original, however this tiny difference in initial conditions had given him completely different results.

This sensitivity to initial conditions became known as The Butterfly Effect. This is because it compares to the idea that a butterfly flapping its wings could produce a tiny change in the atmosphere, and trigger a series of changes which could eventually lead to a hurricane on the other side of the world. The Butterfly Effect is a key characteristic to a chaotic system.

B. LOGISTIC FUNCTION

In this section, I will show the behaviour of a very simple function known as Logistic function. The use of this function is fairly consistent in literature for the introduction to chaos theory. The reason for this is that it is the simplest one dimensional, nonlinear (x squared term), single parameter (b) model that shows an amazing variety of dynamical response. This function can be intuitively seen as a simulation of the population dynamics of some species over the time.

Logistic Function: f(x) = b*x*(1-x) = b*x – b*x^2, where b is a constant known as the effective growth rate.

The population size, ( f^n(x), composition of the function f ‘n’ times) at the nth year, is defined relative to the maximum population size the ecosystem can sustain and is therefore a number between 0 and 1. The parameter b is also restricted between 0 and 4 to keep the system bounded and therefore the model to make physical sense.

The aim is to know, what happens to the long term behavior of the system, for fixed value of the parameter b and any given initial condition, say, x0. So, I change the value of ‘b’ and show the various kinds of dynamical responses of the function.

(a) 0 <= b <= 1
It is clear that for values of b between [0,1], if we start iterating the equation with any value of x the value of x will settle down to 0. The point zero is called the fixed point of the system and is stable for b = [0,1]. A stable fixed point has the property that the points near it are moved even closer to the fixed point under the iterations. For an unstable fixed point the points move away as time progresses. The behaviour of logistic function for b=0.9 is shown on the graph below:

(b) 1 < b < 3

For values of b between (1,3) the iterates instead of being attracted to zero, get attracted to a different fixed point other than zero. The fixed point x = 0 is unstable for b between (1,3) and a new fixed point exists for b >= 1. The new fixed point is that x = 1 - 1/b. The behaviour of logistic function for b=2.8 is shown on the graph below:






(c) 3 <= b <~ 3.5

Apart from 0 and (1 - 1/b) , for all other initial values, the logistic equation will never converge to any fixed point. Instead the system settles down to a period 2 limit cycle. That is, the iterates of the logistic equation oscillate between two values. The fixed points 0 and 1 - 1/b still exist but they are unstable. The behaviour of logistic function for b=3.2 is shown on the graph below:






(d) 3.5 <~ b <~ 3.65

On further increasing b the period 2 limit cycle becomes unstable and a period 4 limit cycle is created. The behaviour of logistic function for b=3.52 is shown on the graph below:











(e) b=4

For b=4, it is found that there are orbits of all periods. This is known as chaotic behaviour. The behaviour of logistic function for b=4 is shown on the graph below:










C. BIFURCATION DIAGRAM

The behaviour of logistic function with different values of ‘b’ can be shown very elegantly on a diagram known as bifurcation diagram. On this diagram, the limiting value of the logistic function is plotted against the value of ‘b’.









D. FORMAL DEFINITION OF CHAOS

Before I give the formal definition of chaos, I will define three terms.

Definition 1 (Density) – A set A is dense in another set B, if for all ε >0 and b Є B there is an a Є A such that | a-b | < ε.

Intuitively, Definition 2 (Sensitive dependence on initial conditions) – We say that the one dimensional map f: D -->D has a sensitive dependence on initial conditions if there exists M>0 such that, for any x Є D and any ε>0 there exists y ЄD and n ЄN such that |x-y| < ε and |f^n(x) – f^n(y)| >M.

Intuitively, Definition 3 (Topological transitivity) - f: D --> D is said to be topologically transitive if for any ε>0 and x,y Є D, there exists z Є D and k Є N such that |x-z| < ε and |y – f^k(z)| < ε. Intuitively, Defination of Chaos:

Let f: D --> D be a one dimensional map. We say that f is chaotic if
1. The set P(f) of periodic points is dense in D.
2. f has sensitive dependence on initial conditions.
3. f is topologically transitive.

Summary of the Papers “The Statue and Clay” By Judith Jarvis Thomson and “Things, stuffs and coincidence” by Nikos Psarros

Both the papers are attempting a solution to the problem of coinciding objects. Thomson takes an example of a piece of clay called “CLAY” and a statue of clay of King Alfred called “ALFRED”. The problem of coinciding objects is about the relationship between CLAY and ALFRED. Both the authors present solutions which are different but still agree on the point that CLAY and ALFRED should not be given same ontologically status i.e. both the authors reject the identity thesis which claims that CLAY and ALFRED are the same.

The Statue and Clay
If I bought ten pound of clay at 9 AM (CLAY) and made a statue at 2 PM on the table (ALFRED) then
Is ALFRED = CLAY?
At 9 AM there was only CLAY and no ALFRED but at 2 PM there is CLAY as well as ALFRED occupying the same place at the same time. Moreover, if I break the statue at 5 PM then again there will be just CLAY and no ALFRED. So, can we say that being a statue is just a temporary property of CLAY? This thesis could be applied to any artifact. This gives an answer to the problem of Identity Thesis. We can say that CLAY = ALFRED at 2 PM and that CLAY ≠ ALFRED at 9 AM and 5 PM. The temporary property of being an artifact can be instantiated from time to time.

But author points to a stronger argument against Identity Thesis. It is the replacement argument. Suppose, I replace one hand of ALFRED with a new one and place the old one on the floor. Then surely, the CLAY is not wholly on the table but it is can be accepted that ALFRED is still on the table because in ordinary thought artifacts are generally capable of undergoing replacement. If we accept this then we have to face a paradox called poor man paradox. If a poor man is given a penny that doesn’t change his status of poor man and he is given another penny that also doesn’t change the situation, but if he is given pennies one by one, then ultimately he would become rich. So how many pennies are needed to make him rich”. Similarly, if all a large part of ALFRED is replaced, then we will have to accept that the old ALRED has been replaced. In such a situation ordinary thought does not supply with answer.

Thomson accepts that the difficulties are serious. “Some philosophers therefore conclude that artifacts cannot undergo replacement of any part, and others that there are no artifacts at all” (Thomson, 1998, pp. 153). Thomson rejects both the arguments and simply supposes that ALFRED is on the table after replacement of one hand but CLAY is not. Now she goes on to define what “constituting is”. The two place relation ‘x constitutes y’ is a temporary relation because before replacement CLAY constituted ALFRED but not after the replacement. This can be easily overcome by using a three place relation instead of two place relation. The three place relation ‘x constitutes y at t’ is a permanent relation.

Now, Thomson establishes a logical framework for further arguments:
(i) x exists at t --> x is a part of x at t.
(ii) x is part of y at t --> x and y both exist at t.
(iii) x is part of y at t <-- --> the space occupied by x at t is part of the place occupied by y at t.

Now, she defines the phrase ‘x constitutes y at t’ or ‘CLAY constitutes ALFRED at t’ by giving logical statements. The first statement in the definition essentially asserts that both CLAY and ALFRED are part of each other. Since we want CLAY to constitute ALFRED, we should not have that ALFRED also constitutes CLAY. The next two statements of the definition endow CLAY with a property that ALFRED does not have. So that we can have CLAY constitutes ALFRED and ALFRED does not constitute CLAY. The second statement of the definition says that CLAY has a part that CLAY cannot loose but which ALFRED can loose. The third statement says that ALFRED does not have any such part which it cannot loose and which CLAY can loose.
So the second and third statements of the definition establish that CLAY is more tightly tied to its parts than ALFRED is. So this condition establishes an ontological difference between CLAY and ALFRED which makes us conclude that CLAY is not identical to ALFRED but merely constitutes ALFRED.


Things, stuffs and coincidence
Psarros approaches the problem from “language-analytical point of view”. He believes that the problem arises when a piece of bronze and a bronze statue are given same ontological status. According to him, “bronze is an abstract substance that does not refer to a thing, but merely to a specific way of talking about a common substantial aspect of things like a bronze statue and a bronze bar” (Psarros, 2001, pp. 27).

The following figure shows his approach:



According to him, the common aspect of bronze bar and bronze statue (the particulars) is called bronze which is abstract and thus a universal. We can proceed further by calling bronze and iron as particulars and the common aspect between them as substance which is a universal. In this way, he makes the relation between bronze and bronze statue clear.

Conclusion
In both the papers, the authors point that the problem of coinciding objects arises when we give same ontological status to CLAY and ALFRED or Bronze and Bronze bar. Further, both the authors give solutions which are quite similar. Thomson gives CLAY a status higher than ALFRED and shows that CLAY has a part that CLAY cannot loose but which ALFRED can loose, which is quite similar to the solution given by Psarros in which he gives bronze the status of genus or universal with respect to bronze statue or bronze bar.


References:

1. Thomson, J.J. (1998), “The statue and the clay”, NOUS, 32:2, (1998), pp. 149-157.
2. Psarros, N. (2001), “Things, stuffs and coincidence”, HYLE--International Journal for Philosophy of Chemistry, Vol. 7, No. 1 , pp. 23-29.
3. http://setis.library.usyd.edu.au/stanford/entries/temporal-parts/.