Logic in Zeno, St Anselm, Occam, Descartes| Francisgevers

Introduction

The aim of this page is not to write yet another manual on logic: many good ones exist already.

Great thinkers have used logic reasoning as a tool since the days of the first philosophers. Their reasoning has not always been flawless: one can mention the sentence “cogito ergo sum” by Descartes; the ontological proof of God’s existence by St Anselm; or the paradox of Achilles and the tortoise by Zeno of Elea. Finally the principle known as “Occam’s razor” has always been correct in practice but, as far as I know, has never been logically explained.

Below are a few articles on these subjects as follows:

Zeno’s paradoxes: Achilles and the tortoise, the arrow

St Anselm’s ontological proof of God’s existence

Occam’s razor

Cogito ergo sum

Finally we will draw a general conclusion.

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Two paradoxes by Zeno: "Achilles and the tortoise" and "the arrow"

The Greek philosopher Parmenides, who lived at some time between the end of the 6th and the middle of the 5th centuries BC, formulated the theory that in nature nothing moves. For him everything is motionless and frozen. He had reached that surprising conclusion through purely theoretical reasoning. He was not concerned by the fact that his conclusion did not correspond with reality. For him, if facts contradicted a conclusion reached by logical reasoning, the facts were wrong, not the reasoning.

His pupil Zeno of Elea developed several paradoxes to support his master’s thinking. The most famous is the paradox of Achilles and the tortoise.

Suppose that the Greek hero Achilles runs a race with a tortoise allowing the reptile a head start. Zeno states that Achilles will never catch up with the animal. Indeed, when Achilles reaches the starting point of the tortoise, the tortoise has progressed and is still ahead of him. When Achilles reaches its new position, the tortoise will have advanced again and will still be ahead of him. This reasoning can be repeated ad infinitum: each time Achilles reaches the previous position of the tortoise it has already progressed further. Zeno concludes that Achilles will never catch up with the tortoise.

Zeno developed a variant of this paradox, based on the same principle. This time Achilles wants to run a distance of, say, 100 metres. He must first run half of that distance (50 metres), then half of the remaining distance (25 metres), then again half of the remaining distance (12.5 metres) and so on ad infinitum. Zeno concludes that Achilles will never reach the end of the 100 metres.

In modern mathematics these two paradoxes are solved by using the theory of convergent series, which was of course not yet developed in Parmenides’ and Zeno’s time. It shows that the sum of an infinite number of numbers does not necessarily tend to infinity but can, as in this case, converge towards a limited total. These formulae allow calculation of the exact moment when Achilles will catch up with the tortoise. But they also contain errors of pure logic which could have been detected in his days.

In his “Physics” Aristotle points at a first error by noting that if the distances decrease the times needed to cover them decrease as well. However this argument does not suffice to disprove the paradox. Even if the time intervals become shorter and shorter, their sum could theoretically still tend to infinity. But Aristotle’s argument can be generalized by stating that Zeno uses distance and time in different ways without justification. His paradox is based on the implicit assumption that any distance can be divided into an infinite number of segments, but that time cannot. He does not justify this difference which he uses as a hidden axiom. His paradox only works if one accepts that difference.

There is also a second mistake. In both paradoxes, the one where Achilles competes with the tortoise and the one were he runs hundred meters, Zeno reasons only about distance. Logically this should lead to a conclusion in distance as well. But that is not what Zeno does: he jumps to a conclusion in time by saying “never”. To draw a conclusion concerning time, Zeno should have established a link between distance and time but he did not do that.

Both paradoxes contain the same two mistakes: they handle distance and time in a different way without justification; and they draw a conclusion in time after reasoning in distance, again without justification.

Another of Zeno’s paradoxes about motion is known as the paradox of the arrow. To fly, an arrow must obviously change its position. Zeno notices that at every stage of its flight the arrow is in a precise position, occupying exactly its own volume. Such a moment has by definition no duration, in modern mathematical parlance it is a dimensionless point on the timeline. During that dimensionless instant the arrow is necessarily motionless. This is true for every position the arrow occupies during its flight: it is always motionless. Zeno concludes that it cannot fly.

This paradox is more difficult to unravel. The difficulty is the mathematical definition of point, line and plane. A point has no dimensions. A line segment has only one dimension: its length, but no width or height, and a surface has length and width but no height. How can an infinite series of dimensionless points create a line segment that has a dimension: its length? And how can an infinite series of lines without width create a plane? In other words, what is the result of infinite times zero or ∞x0? The rule in mathematics is that the result of this multiplication is undetermined.

The result will be different in different cases There is no general solution. In the case of a particular segment of line, the result of ∞x0 is the length of that segment. This looks very much like a mathematical trick, doesn’t it? Yes possibly, but there is no way out of it. Here again the problem is one of basic axioms. Zeno pretends that an infinite times zero remains zero, in other words that ∞x0=0. To disprove the paradox, one must accept that ∞x0 is undetermined. The scientist will say that the arrow flies which proves that ∞x0 is undetermined. But again, the problem can only be resolved by looking at the underlying axioms.

To build a reasoned argument, one needs to start from a number of axioms or hypotheses. By definition these will not have been proven. The result of every reasoning depends on the chosen axioms. That is why scientists observe nature to verify the validity of their axioms. But this obligation to observe nature is also an axiom, not accepted by Parmenides and Zeno.

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The error in St Anselm’s "ontological" argument

Why his argument does not prove the existence of God

St Anselm was born in Aosta (Italy) in 1033. He lived first in the Benedictine abbey of Bec in Normandy, before moving to Canterburywhere he was archbishop from 1093 until his death in 1109. He composed two theological books, the Proslogion and the Monologion. His greatest claim to fame is the ontological argument of the existence of God found in the Proslogion. The Fordham University of New York translates the argument as follows (I have replaced their slightly obscure sentence “something greater than which cannot be thought” by “something than which nothing greater can be thought”):

And indeed we believe you are something than which nothing greater can be thought. Or is there no such kind of thing, for the fool said in his heart, “there is no God”? But certainly that same fool, having heard what I just said, “something than which nothing greater can be thought”, understands what he heard, and what he understands is in his thought, even if he does not think it exists. For it is one thing for something to exist in a person’s thought and quite another for the person to think that thing exists. Thus even the fool is compelled to grant that something than which nothing greater can be thought exists in thought, because he understands what he hears, and whatever is understood exists in thought. And certainly that greater than which cannot be thought cannot exist only in thought, for if it exists only in thought it could also be thought of as existing in reality as well, which is greater. If, therefore, that than which nothing greater can be thought exists in thought alone, then that than which nothing greater can be thought turns out to be that than which something greater actually can be thought, but that is obviously impossible. Therefore something than which nothing greater can be thought undoubtedly exists both in thought and in reality.

Let us try to reword this slightly obscure medieval text:

“Consider that the term God means the most perfect thing that could be thought of. The fool (in Anselm’s terminology, the atheist) hears this definition and understands it, even if for him this thing does not exist. This definition is present in his mind. The concept “the most perfect thing that can be thought of” is present in every mind, even the mind of the atheist who refutes the existence of God. However, a God who exists is more perfect than a God who does not exist. If he did not exist there would be a contradiction: the most perfect thing that is present in the minds would not be the most perfect. The only way to take away this contradiction is that God exists.”

This argument is brilliant. For it to work, the definition must not only be in the mind of the atheist; he must also understand it. Indeed, an idea that is not understandable can’t be a valid proof of anything.

From the outset this argument has been much criticized but it has also been used by some scholastics and even by Descartes. Today most people, believers and non-believers, sense that there is something suspect about the argument. But where is the error?

Still in St Anselm's lifetime the Benedictine monk Gaunilo of Marmoutiers objected to this argument. He suggested that it would suffice to think of the most perfect island for it to exist in the sea. Anselm’s reply is as clever as his proof. An island is necessarily limited in its perfection and will therefore always lack many qualities; so will the imagined island. It will always be possible to imagine another island, more beautiful than the previous one. The sentence “something than which nothing greater can be thought” does not apply to anything like an island. The argument only holds for something that is totally perfect. Absolute perfection is an attribute of God only, and the argument only works for him.

Several philosophers have commented on the argument. Amongst them Kant has remarked that a thing that exists is not necessarily more perfect than a thing that does not exist. Surprisingly most thinkers consider that this argument is not enough to disprove Anselm’s position. We will come back to this point.

Most philosophers agree that analysing the meaning of a word does not suffice to prove the existence of something in the real world. Still, the exact mistake in Anselm’s argument does not seem to have been identified.

Let us start with an analysis of Kant’s objection. To prove something one has always to begin by establishing a set of axioms or hypotheses. By definition, these will not have been proven. The quality of any argument will depend on the quality of the axioms on which it is based. St Anselm states that something that exists is more perfect than something that does not exist. This statement has not been proved and is an axiom. By changing this axiom, one could prove exactly the contrary. One could declare that something that does not exist and has therefore been idealised in the mind is necessarily more perfect than something that exists; and conclude that this proves that God does not exist. This proof would again only be valid based on the underlying assumption. No proof is ever absolute. By choosing the basic axioms carefully it is theoretically possible to prove the most absurd things.

In his reasoning St Anselm also leaves a gap. He starts by giving a general definition of God: “something than which nothing greater can be thought”. This definition is self-contained and there is no need to describe further the qualities of such a completely perfect thing. He then changes his method and quotes one of the qualities this perfect thing should have: existence. He jumps from a theoretical definition to a description of qualities. What about the other qualities of this God that he now describes? Even believers admit that there are contradictions in the qualities of an absolutely perfect God. How for instance could he be perfectly just whilst being at the same time perfectly merciful? For Anselm’s proof to work, “that same fool, having heard what I just said … understands what he heard, and what he understands is in his thought”. In other words, the description of the perfect qualities of God must be in his mind and he must understand them. Nobody, either believer or non-believer, with his limited brain capacity, can conceive a complete set of “perfect qualities”.

Either we stick to the general definition without mentioning any of the qualities and consequently not existence either. Or we mention one of the qualities and must envisage the others which is impossible. St Anselm’s proof works in neither of these two cases. As revealed in Anselm’s response to Gaunilo’s remark, the argument is only correct for an absolutely and completely perfect God. It is ridiculous to imagine that a limited human mind can understand infinity. The given definition cannot be understood by human minds, whatever Anselm may say. Neither the believer nor the fool “understands what he heard”. Anselm’s definition can perhaps exist in the fool’s mind, but he can not understand it.

The above reasoning can be worded differently. To reply to Gaunilo’s objection, Anselm had to reformulate his starting principle. Existence had to be the only missing quality. Having initially selected the definition “something than which nothing greater can be thought” he had to change that into “something perfect”. He uses as starting point the fact that perfection exists, at least in thought. In other words, his starting implies already that God exists .

A last thought. Can one apply any apparently correct reasoning to the real world? Philosophers can be divided into two opposing camps: rationalists and empiricists. For rationalists, truth can only be obtained through correct reasoning whilst for empiricists every thought must be measured against nature. Who is right? Philosophers have argued about this question since Parmenides. For the empiricist, Anselm’s conclusion should be verified in the real world. This is certainly not what Anselm intended. Anselm is a rationalist for whom truth is to be found through reasoning, not through testing. But from the rationalist’s perspective, the proof doesn’t work either for he bases it on the starting hypothesis that perfection exists. Every proof is based on start axioms. No proof is ever absolute, whatever Anselm may think.

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OCCAM’S RAZOR

An Explanation

« Entia non sunt multiplicanda praeter necessitatem »

William of Ockham or Occam (c.1285-1349) was an English Franciscan monk. He was a philosopher and a theologian. His theories were about to be denounced by the Pope (he was eventually excommunicated) when he fled to Munich where he ended his days as a scholar. His most famous contribution to philosophy is the well-known “Occam’s Razor” described here.

Literally the above statement means “entities must not be multiplied beyond necessity”. This has become one of the fundamental scientific principles, often paraphrased as : “If two theories explain a phenomenon equally well, the one which contains fewer entities should be selected”. In other words choose the simplest one. Occam’s razor is so called because it advocates the “shaving” of all needless complexities.

In everyday life this principle seems obvious. Why take a detour when you can go straight to your destination? In the field of science the Razor principle is systematically applied even though a rational explanation has not yet been found. Indeed a priori there is no reason why nature should always be simple. Nevertheless until now the principle has never been proven wrong.

19th century scientists believed that there would soon be no need for scientific research as everything would have been “explained”. They even advised young university students not to study physics, a branch with no future. Nobody believes this now. Not only does ultimate truth seem to have become more remote with the ever-increasing number of discoveries made by science, but today most scholars admit that the purpose of science is not to explain, but to describe natural phenomena. Science does not answer the question “why” but “how”. In electricity, for instance, experiments show that the plus and the minus attract each other, while two positives and two negatives repel each other. Science doesn’t explain why things happen like this or why it doesn’t happen the other way round. It states the fact; that is all.

However, I would even question this principle of description. Is that really what science does? Here are some examples of what I mean.

The ideal gas law states that pV=nRT. In this formula p represents pressure, V the concerned volume, n the quantity of gas, T the temperature and R is a constant. Even if it is approximate as gases are not “ideal”, the formula has the advantage of being simple. It shows, for example, that if gas is heated at a constant volume, the pressure will increase. It provides practical calculations for modern appliances such as the manufacture of fridges.

In fact this formula does not at all describe what is happening. In reality huge quantities of gas molecules travel at great speed, crash into each other and bang against the surfaces of the container. Pressure is therefore not the same everywhere. A sufficiently small appliance would measure higher pressure at the point where a molecule hits a surface and no pressure at all beside this point. Our large measuring equipment merely records the mean value between all these impacts. There is no link between the formula and the complex movement of molecules. The aim of the formula is moreover not to describe the reality, but to provide a simple and practical method of calculation. It would be impossible to translate into a formula the performance of billions of molecules which move and bang against each

other. As Occam’s razor recommends, the simplest formula (giving results which are satisfactory) will therefore be adequate. The fact that this formula does not describe the phenomenon is irrelevant.

In this example we have looked at a natural phenomenon that can be described qualitatively (the movement of molecules), but which is too complex to be translated into an equation. Science makes do with creating a simple and approximate formula which is enough for most practical applications.

However in most instances the scientist has no idea what is really happening in nature and is limited to finding workable formulae and models. For example, the atmosphere is criss-crossed in all directions by what we call electromagnetic radiation, also known as electromagnetic “waves”. There are radar, light, radio and television waves and many more. Each space, however tiny it may be, is criss-crossed by all sorts of radiation from every direction. These waves are constantly passing through our bodies without us noticing anything.

Why do we use the term “waves” for radiation? To define the concept of waves, the mathematician quotes the examples of the circular ripples which appear when a stone is dropped into a perfectly still lake or of sound which propagates in a longitudinal wave. No problem. Everyone can visualize these waves easily. These types of waves have prompted mathematical formulae. It appears that these formulae are equally effective in calculations for electromagnetic radiation. Their use makes possible the manufacture of the many appliances which surround us:

television, radio, telephone, radar, etc. But does that mean that what we call magnetic radiation are actually “waves” just because the use of these formulae gives good results? Nothing is less certain. How would these electromagnetic waves move in empty space? This problem led 19th century scientists to invent ether, a hypothetical omnipresent substance, which was said to be the essential medium for their transmission. They suggested that electromagnetic waves move in ether as the ripples of water undulate on the surface of a lake disturbed by the throwing of a stone or as sound moves forward in air. This hypothesis was abandoned at the beginning of the 20th century.

In addition, these formulae are not sufficient to explain all the phenomena associated with electromagnetic waves; other formulae had to be created to deal with these phenomena. The new formulae consider electromagnetic

radiation as corpuscular and constituted by what are known as photons. Thus there are two groups of contradictory formulae. Circumstances dictate which one should best be used. We can safely conclude that neither describes what is happening in reality.

Another example is gravity. Gravity is described by simple formulae which can calculate the movements of solid bodies. But what does gravity consist of? How is it that it acts instantaneously with infinite speed? Is it a wave? We know nothing about it. We are quite unable to “describe” gravity. Once again science merely presents formulae which enable practical applications without describing the phenomena. Gravity is no more than a name. The formula which states that the force of attraction between two objects is proportional to their mass and inversely proportional to the square of the distance that separates them does not describe the nature of the phenomenon.

It would be possible to quote numerous examples, but we will leave it at that.

Science is only rarely able to describe the natural phenomena; in most cases it makes do – not being able to do better – with establishing formulae or models allowing the exploitation and best use of natural phenomena. A typical example can be found with the theories of Einstein. They are an improvement of Newton’s formulas. In daily life however the differences produced by Einstein’s theory are so small that they don’t matter at all. Very wisely, and as proposes Occam’s razor, we continue to use Newton’s formulas and only use Einstein’s when the differences become noticeable.

Occam’s razor is simply a rule of reason in a scientific world which must be satisfied in putting forward mathematical formulae, being rarely able to “describe” the phenomena.

But it is possible to draw more conclusions from the foregoing. In many areas which are qualitatively completely unknown to us, brilliant scholars have succeeded in establishing mathematical models and principles which are confirmed by measurements and tests. This is the case for instance in the area of the very large, the cosmos, and of the very small, the structure of the atom where direct observation is impossible. Does this allow us to conclude that these formulae and models describe what happens in reality? When a new method of calculation allows the prediction of phenomena which were previously unpredictable, can we conclude that these mathematical models are exact descriptions of what is happening in nature? Many scientists make that assumption which is however not logical, as we discussed above.

I would suggest a particularly bold example, but one that in my opinion is justified on the basis of my previous argument.

The theory of the Big Bang allows us to make many calculations which can be verified. However that does not prove that the Big Bang has really taken place, in the same way as the formula pV=nRT does not describe the real behaviour of gasses; nor does the formula used for electromagnetic radiation allow us to conclude that it is a wave.

The time has come for science to recognise its correct place and abandon the illusion of being able to “describe” nature by means of mathematical formulae, even those which are proven. Nature can only be described by direct observation; and not by mathematical formulae which remain useful but non-descriptive calculating tools.

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I think, therefore I am

René Descartes (1596-1650) is considered to be the father of modern philosophy. Not only was he a great philosopher, he was also a mathematician of genius who created analytical geometry and was a precursor to Newton and Leibniz in infinitesimal calculus. He is probably best known for his reasoning “I think therefore I am” which, surprisingly for a genius of his level, is tautology. Originally he wrote this sentence in French in his “Discourse on Method” in 1637 and only translated it later into the famous Latin phrase “Cogito ergo sum”.

His aim was to prove that God existed starting from nothing. He decided to discard with everything and adopted the principle “I doubt everything”. He concludes that there is therefore one thing he is certain of: he is certain that he thinks. He then continues with his famous reasoning “I think, therefore I am”. However his starting point is already flawed: he doesn’t doubt everything as he knows that he doubts. If he had doubted everything, he would even have doubted his doubt.

His conclusion is “I am”, in other words “I exist”. From the outset he uses the word “I” for he says “I” doubt and “I” think. His starting point already implies his existence. In the end all he says is “I am, therefore I am” or “I exist, therefore I exist” which is correct but a tautology.

His main mistake is that he wanted to prove something (in this instance that God exists) starting from nothing. It is impossible to prove anything starting from nothing. To build an argument one needs to start with a few initial building stones, i.e. axioms or hypotheses. Often these axioms are implicit and are not declared as such. But they are there, otherwise it would be impossible to build up the argument. Descartes’ “I am” or “I exist” is in fact one of his axioms.

Descartes knew what he wanted to prove beforehand. All he wanted to do was to build a reasoning that would lead to the conclusion he wanted to reach, in this instance God’s existence. As is demonstrated here, it is always dangerous to fix your conclusions first and then to compose your argument on the basis of the results you want to obtain.

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General Conclusion

All reasoning is based on a certain number of axioms which – by definition – have not been proven. Often these axioms are not explicitly mentioned as they are considered self evident.

It is possible to prove anything and everything by changing axioms. When analyzing a reasoning one should always try to analyse which axioms form its basis. Very often if the conclusions seem eccentric they will be based on eccentric axioms. But it is impossible to prove that they are: one can only propose another axiom which doesn’t prove anything.

Every scientist and every empirical philosopher uses the basic axiom that nature is the measure of all things. This axiom seems self evident. It has allowed the construction of cars, pressure cookers and nuclear bombs. But in the end it is only an axiom which many philosophers do not accept.

This leads us back to the traditional opposition between rationalist and empirical philosophers: the former maintain that reason is the only source of truth, the latter that observation of nature through the senses is the only solution. If the rationalists were right, this would mean that truth does not exist as it would depend on the whimsical choice of personal axioms. I therefore remain a staunch empiricist.

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