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.