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Precisely what is wrong with Zeno’s Achilles and the Tortoise argument?

Only some 200 words of Zeno survive. We rely on later commentators such as Aristotle and Simplicius. The latter first called Achilles’ opponent the Tortoise.

Zeno was a pupil of Parmenides, and the 4 paradoxes of motion (Achilles, Dichotomy, Arrow, Stadium) attempt to show that motion is impossible in line with the nothing-changes view taught by the great man.

Since motion clearly is possible, indeed actual, there must be something wrong with the arguments as you suggest.

I will briefly outline the Achilles and the Dichotomy, which are logically equivalent, and then suggest how they might be refuted.

The Achilles:

Achilles (A) is a good runner. He sportingly gives his slower rival, the Tortoise (T), a start. The race begins. By the time A reaches T’s start point, T has moved on to a new point. By the time A reaches that new point, T has moves again to a further point. By the time A reaches that further point, T has again moved ahead, and so on endlessly. A can never catch T.

The Dichotomy:

Version 1: To travel any distance, I must first reach the halfway point. Then I must reach the halfway point of the remainder, then the halfway point of the new remainder, and so on endlessly. I can never complete the journey.

Version 2: To travel any distance, I must first cover half the distance. To do this I first have to travel half of that half (first 1/4 ). Before that, half of that quarter (first 1/8), before that, 1/16, and so on endlessly. I can never start the journey.

The paradox is not that we must travel an infinite distance, or for infinite time. Clearly, knowing the speeds of A and T, and the length of T’s start we can easily calculate where/ when A catches T, or when the Dichotomy runner completes the run. The paradox is that an infinite number of actions (tasks) seems necessary — A has to pass every one of the unending sequence of points where T once was.

What’s wrong?

To refute the argument we must deny at least one of its 3 presuppositions, which are:

1. In travelling a distance we must cross each and all of the intervening points.
2. A line consists of an infinity of points.
3. It is impossible to complete an infinite series of actions (tasks).

Aristotle denied 1., saying a line can’t consist of points, they have no size, whereas a line has. A point is potential, becoming actual only if we divide the line there. The paradox invites us to repeatedly divide the line at an infinity of potential points, but A does not have to touch an infinity of actual points to catch T.

Others deny 2., saying a line doesn’t consist of an infinity of points, rather space is not continuous but consists of tiny discrete units (quantized). In covering a distance we traverse a finite number of  space quanta. Motion is jerky but this is undetectable due to the fantastically tiny size of the quanta.

Yet others deny 3., saying that an infinity of tasks is possible. This is a subtle business, bringing in Aristotle and Dedekind cuts, and dealing with it would make this answer too long (ask me if you’re interested).

After 2500 years there is still debate, and no closure, especially about 2. and 3.

Some modern mathematicians offer as a solution that that the time/ distance till A catches T is the (finite) limit of an infinite convergent series (1/2 +1/4 +1/8 + etc). But this simply tells us what Zeno already said, that the distance is finite, just infinitely divisible, and doesn’t explain how we complete the task.

I hope this helps.

I am 65 years old and still struggling with philosophy.

I started with Father Copleston’s volumes on The History of Philosophy — struggled.

Thinking about taking a course with the University of Edinburgh — time is running out to decide. Please help me with a book that will help me to understand and help me progress with other books or courses. Please help!!

Every student starting out with philosophy is different, so I am not going to recommend a specific book for you to read. Age is not an issue. (Craig Skinner , one of the members of the Ask a Philosopher panel, gained his Philosophy BA in his 70s.) However, you could start by looking at our Pathways to Philosophy Introductory Book List. Use the comments as a guide and look up the reviews on Amazon — just to get a sense of a selection of books you might like.

I have never attempted to read Copleston, although this evening I did look up the article in Wikipedia which states:

“Throughout the eleven volumes Copleston’s Roman Catholic (Thomist) point of view is never hidden. All the same, it seems generally accepted that Copleston’s treatment is fair and complete, even for philosophical positions that he does not support.”

It’s not that I have anything against Jesuit Priests. Life is too short. Eleven volumes! But I totally understand the desire to tackle something ‘big’. I started off hunking Benjamin Jowett’s Collected Dialogues of Plato and Kant’s Critique of Pure Reason from Swiss Cottage Library, London.

I didn’t read the tomes. I dipped. I picked things that looked interesting. They gave me a sense of what an incredible, huge subject Philosophy is. Endlessly deep, seemingly impossible to master.

As I seem to recall, the first book I actually read all the way through was A.J. Ayer Language, Truth and Logic. It’s a young man’s book, published when Ayer was just 26. I was 21, so there was a connection there. I’m not saying it wouldn’t be a book for you, but you might see it in a different light. (Still worth a look, though.)

But why even make the effort to tackle a subject, if you know in advance (or fear) that you will be struggling with it? Maybe you won’t be the best philosophy student or get a First Class BA, but would that be so bad? I have interests that I pursue, knowing I will never ‘get good’. But that doesn’t deter me from enjoying the challenge.

Take the University of Edinburgh course. You will not be the only one struggling, I promise. What you will gain is a sense of cameraderie that only philosophy students know. There is nothing better or more worth doing than philosophy. Without philosophy, all the learning in the world would not be worth a damn. (That was said by some famous Greek — I don’t remember who.)

Whatever philosophy book you pick, you will struggle. You had better struggle, because otherwise it’s almost a foregone conclusion that you’re missing something. That’s why books that seem easy to read can sometimes be the hardest. You skate over the ideas, but none of them really grabs you.

As time is short, I would pick a shorter rather than a longer book. Something you can read in a week or two, if you make the effort. — Go for it!

When, if ever, should a terminal patient’s right to be told the truth about their condition — that they are dying — be overridden?

I dont think a patient of sound mind who asks for the truth should be lied to.

As a medic, I would say medical ethics revolves around 4 principles:

• beneficence (do good)
• non-maleficence (do no harm)
• autonomy (patient’s right to truth and to decide)
• justice (fair sharing of limited resources)

Your question relates to autonomy. Relatives often say the patient shouldn’t know the truth because she couldn’t cope with it — she would give up, wouldn’t fight any more. So, they say,  autonomy clashes with doing no harm, the latter should prevail, I should lie. Whilst I always take account of relatives’ views, I am not bound by them, and I have never agreed with this view. It usually means it is the relative who cant cope, won’t know what to say if mum starts talking about death. And if we go along with it, the outcome is bad: the patient soon realizes she is dying anyway, but can’t talk about it because her family don’t want to, and so she is alone and isolated just when she most needs family support. So, I explain all this to the relative, and tell the patient the truth.

Two caveats.

First, a patient’s right to the truth doesn’t mean I have a right to ram the truth down her throat, as it were, when she doesn’t want to talk about it. I am guided by the patient’s wishes. So, having examined her and got back test results, I will start in nonspecific terms — there’s a shadow on your x-ray we need to investigate further, say. Then, “Is there anything you want to ask?”, and now we find out how much the patient wants to know, at this stage anyway. Some say no, that’s fine, lets get on with the tests, others ask if it could be cancer. And so it goes as we investigate further. If it turns out to be a fatal condition for which little can be done, I will say so if asked directly, otherwise will talk of treatment which may help for a time. If a patient has, or probably has, a very serious condition, but one which may be curable, and decides to bury her head in the sand, keep clear of doctors and hope for the best, then I will make sure she knows the serious position even though she did not ask me to spell things out.

Second caveat. Docs know only too well that they often get it wrong, so we must always consider very carefully before pronouncing an illness terminal or being too exact or sure about life expectancy. Second opinions may be a good idea.

I hope this is helpful to you.

In view of Hitler or Trump, by what criteria — if any — would you call a person or its ardent followers sane or insane? Or don’t you see any difference between, say, Jesus and his followers and Hitler and his followers? Are there no valid criteria?

First off, allow me to decline your invitation to share my views about Hitler, President Trump, or Jesus Christ. Except to say that there are readers of this page who are neither ardent or fanatical, who think that Trump is a good guy, and there are also readers of this page who think that Jesus was actually not such a good guy.

Seven decades after the end of WWII it is still politically incorrect to say anything good about Hitler or his followers. Few would doubt that Hitler was evil. But what is really scary is that many of his followers were ordinary people, decent folks who helped their neighbours and were kind to animals.

Sanity/ insanity is a much abused distinction — witness the Soviet treatment of dissidents. Anyone who objects to communism must be insane, mustn’t they? I would be very careful bandying that label!

Charles Manson, who died just a few days ago, would be a better example. Were his followers insane? If the answer to that is, No, then you really couldn’t have a better case. Were they brainwashed? Is being brainwashed a form of insanity? Certainly, if the result is that you become a jibbering idiot (as in the movie The Ipcress Files, 1965). But if it’s just another name for indoctrination, then you would have to call a large proportion of the human population ‘insane’.

Being ‘ardent’ or ‘fanatical’ isn’t a form of insanity. It’s one of the normal ways in which human beings express themselves. Normality is a spectrum. It is normal to be what some would term ‘irrational’ about some person or topic. Having the power of reason doesn’t entail that an individual is particularly good at reasoning, or that they are what you or I would call a ‘reasonable’ person.

And even if you are a ‘reasonable’ person who is ‘good at reasoning’, there may still be particular things — in fact, there probably are — which you are irrational about. (Call me irrational, but I won’t listen to any argument, however persuasive, for the view that Eric Clapton wasn’t the greatest rock guitarist of all time.)

What is the difference between the people who followed Jesus and the people who followed Hitler? Do I have to say? We admire those who preach love, and despise those who preach hatred. But don’t you hate Hitler and the Nazis? At least you can say you hate what they did, even if you are willing to extend to them Christian ‘love’.

If you are looking for criteria for sanity, there’s the Diagnostic and Statistical Manual of Mental Disorders (DSM-5), published by the American Psychiatric Association. Evil isn’t listed in the manual because it isn’t a form of insanity. There’s a lot of evil about. Sad to say, evil and malice are natural human tendencies, a place on the spectrum of normality.

What do these words mean in Kant’s 12 Categories: Inherence and subsistence, and community? These three terms are in the third group of categories (of Relation). I’d be happy with any url that would define and discuss all 12 of Kant’s categories.

The Categories of Relation: Substance (Subsistence), Causality (Inherence) and Interaction (Community) are outlined by Kant in Section 3, Chapter 2, Book 2 of the Critique of Pure Reason.

As written, in Section 3 ‘Systematic Representation of all Synthetical Principles of the Pure Understanding’, the Analogies of Experience are outlined.

Perception is only possible by the continuous connection of Intuitions. Otherwise, they would be isolated, contingent and unconnected. So what connects them? Time cannot, as an a-priori condition of experience, (See the Transcendental Aesthetic of the Critique) be a subject of perception. Here enter the First Analogy.

First Analogy: The Permanence of Substance.

As Time is not an object for perception, there must be something by which it is represented; a something in which change, modification and successive changes of state, occurs. For Kant, this ‘something’ is precisely Substance. Whilst there are changes, modifications etc to objects, their underlying Substance remains unchanged. Otherwise there would be an unconditioned, chaos of perceptions without coherence. For example, the tree exudes a succession of changes: buds become leaves, leaves which fall yet the underlying ‘thinghood’ of the tree remains as a ‘ground’ for such changes.

Second Analogy: The Principle of the Succession of Time according to the Law of Causality.

When a human subject looks at a house, gazes from its roof to its foundations to its windows and so on, there is a succession of intuitions in Time which is not causally determined. This is because they are dependent on that human subject — they are a Subjective sequence of events. It is otherwise when gazing at a boat flowing down a river.

Here, there is a linear, successive sequence of events: A then B, then C, then D and so on which cannot be otherwise. The boat was once high upon the river, then it was adjacent to the perceiver, then it is further down the river. This succession is conditioned by and made objective by the Transcendental Law of Causality. It cannot be otherwise, for then knowledge deriving from linear, successive perceptions would be made impossible – contrary to our actual experience.

Substance underpins the succession of Time according to the Law of Causality.

Third Analogy: Principle of Co-Existence according to the Law of Reciprocity or Community.

All Substances, existing under the conditions of Space and Time, exist in a simultaneous relation of reciprocity: of reciprocal action co-extensive with each other in a plurality of ways hence, – a reciprocal community of substances. Unlike the Law of Causality above where we have A then B then C and so on, here we have A affecting D or B (or any other term affecting any other) and vice versa – simultaneously in Time and Space.

The synthesis of the Understanding thus has objective, Transcendentally conditioned intuitions of a ‘compositum reale‘ – distinct phenomena, all in inclusive connection with each other in various ways such as inhereing with each other, consequences as effects of causes and composition as a manifold of intuitions in unity (or Reciprocal Community), simultaneously experienced by the ‘I think’ as the unity of apperception

Hope this is useful, Marden.

To act morally is it necessary to fight against one’s desires?

This all depends upon what desires one has and what moral background we are comparing them with; but in general, I would say that one does not necessarily have to wrestle with one’s own desires to act morally. About a month ago I answered the question ‘What is a Moral Society?’ and I would reassert part of my answer: that it is a society or at a lower level a community, that sets a shared code of conduct, that is agreed by the majority of its inhabitants, that we may term ‘morality’. As most of the inhabitants of a society would agree upon its values and live their lives by them, then it follows that the seemingly presupposed notion that persons have to restrain themselves to live morally would be fallacious.

I suspect that the majority of persons do not need to restrain themselves in society as they have a desire to be part of a community, or in other words a sense of belonging, which outweighs any other desires to better oneself through antisocial acts that harm others. Additionally, one’s conscience would ensure that one’s behaviour aligns itself with the prevailing morality. Either reason may be enough to ensure that most individuals behave themselves but the combination of both ensures all but the most errant individuals cannot be considered to be moral actors.

With regard to persons needing a sense of belonging, this may be a facet that has evolved as living in social groups has been beneficial to humanity’s survival; and without it society would undoubtedly crumble. I would think that there is a deep-seated, innate desire to belong to groups and this entails absorbing one’s community’s standards. Hence, most persons’ desires would coincide with others’.

Accompanying this, most persons also have a conscience and this facet is often called into play when persons are tempted to commit antisocial acts. In fact society exercises so much disapproval over persons who do not believe or adhere to its standards that this phenomenon alone may guide persons’ actions and ensure compliant behaviour; although disincentives are also provided by punishment established through criminal justice systems. Hence, persons with antisocial desires often align their behaviour with moral standards; and the more they align their behaviour the easier this process seemingly becomes.

The commonest schools of philosophy have realised that persons wish to live lives attuned to their society’s standards. Virtue ethicists wish to channel this phenomenon by educating persons from a young age to behave compliantly and rely upon heightening a person’s sense of belonging to ensure this; whilst deontologists or utilitarians may set boundaries by which persons actions may be judged and therefore seemingly place more reliance upon invoking a person’s conscience.

Hence, I would conclude that for the majority of persons, for the majority of the time, they do not have to restrain their desires as they either coincide with society or become aligned. Furthermore, in general, all persons in a society are judged by the same standards although there are times when the logic fails: for instance, those with prestige, talent or even illness may be offered more latitude when they are judged by others.

As a coda, I would add that there are always those who cannot comply with society’s morality and often must be punished. Hence, since the times of early Greek philosophy, much debate has occurred as to why persons intentionally refrain from acting morally and for further reading, one may seek entries in philosophical dictionaries concerning the conundrum of akrasia.

### Are you a philosophizer?

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