By Peter Boghossian and James Lindsay

If you want to be a good philosopher, don’t rely on intuition or comfort. Study maths and science. They’ll allow you access the best methods we have for knowing the world while teaching you to think clearly and analytically. Mathematics is the philosophical language nature prefers, and science is the only truly effective means we have for connecting our philosophy to reality. Thus maths and science are crucial for good philosophy – for getting things right.

Truth is not always intuitive or comfortable. As a quirk of our base-ten number system, for example, the number 0.999…, the one that is an infinite concatenation of nines, happens to equal 1. That is, 0.999… *is* 1, and the two expressions, 0.999… and 1, are simply two ways to express the same thing. The proofs of this fact are numerous, easy, and accessible to people without a background in mathematics (the easiest being to add one third, 0.333…, to two thirds, 0.666…, and see what you get). This result isn’t intuitive, and – as anyone who has taught it can attest – not everyone is comfortable with it at first blush.

The sciences, which were largely born out of philosophy, are also replete with nonintuitive, and even uncomfortable truths. The most extreme examples of this are found in quantum mechanics, with interpretations of double slit experiments, quantum entanglement, and the Heisenberg Uncertainty Principle confounding essentially everyone. But even sciences investigating scales more familiar to us, like biological evolution, are nonintuitive and uncomfortable to the point of being rejected by surprising numbers of people despite overwhelming scientific consensus spanning nearly a century and a half.

Continue reading by clicking the **name** of the source below.

## 68 comments on “Want to Be Good at Philosophy? Study Maths and Science”

BS!

0.999… is not an actual number; if the decimal goes on infinitely, then it is never actualized. It’s basically a conceptual or an “imaginary” number and can never be equal to 1, an actual, finite number.

I just had to get that off my chest. If I’m wrong about this, I’m eager to have someone explain how.

Report abuse

Hi PeacePecan,

Thanks for reading. I hope I can add some clarity to the relevant concepts in play regarding 0.999… so that you might better understand. 0.999… is a decimal expansion of a number. A decimal expansion is the way that we write down the value of a number using the various powers of ten as a base. Some other examples of numbers, which you call “actual, finite numbers,” written in decimal expansion are 352 (which is 300+50+2, or 3 of the second power of ten, 5 of the first power of ten, and 2 of the zeroth power of ten) and 0.25 (which is 0.2+0.05, or 2 of the negative-first power of ten and 5 of the negative-second power of ten). Every real number has a decimal expansion, and this seems to be where you have gotten lost.

The set of real numbers can be defined as “the set of numbers with a decimal expansion,” but this isn’t very helpful since that’s what we want to know about. Another definition for that set that’s more accurate is the set of rational numbers (those that can be written as a fraction of integers, which are whole numbers, positive, negative and zero) and the irrational numbers (those real numbers that cannot be written as a fraction of integers). Again, this characterization isn’t obviously helpful, but with a little digging it is. If you take all of the numbers that have a terminating decimal expansion (which you mistakenly called “actual, finite numbers”), the set of real numbers is the complete set of numbers to which any convergent sequence of rational numbers can converge. Imagine the number pi, 3.14159…. Pi is the value of convergence of a sequence of rational numbers that begins (3, 3.1, 3.14, 3.141, 3.1415, …). (There is a formal definition of the real numbers that goes pretty far outside what I can briefly explain here — the totally ordered, Dedekind-complete field under the usual addition and multiplication — or, equivalently, the rational numbers together with every point of convergence of any Cauchy sequence of rational numbers.)

Some rational numbers, like 1/2, have decimal expansions that terminate (because the denominator has only powers of two and five, which are the prime factors of ten, hence “decimal”). Other rational numbers do not have decimal expansions that terminate, like 1/3, because the value in the denominator cannot be written as a product of powers of 2 and 5. So when you try to place a number like 1/3 onto a number line that has been cut up so that each segment is subdivided into 10 pieces, you will see that 1/3 is to the right of 0.3, 0.33, 0.333, 0.3333, and every terminating decimal that is a concatenation of 3s, but it is to the left of any such sequence 0.333…34.

The only possible resolution to this issue is to give 1/3 an infinitely long decimal expansion, 0.333…. This doesn’t make 1/3 imaginary (if I have three oranges and give you one of them, your share of the whole isn’t imaginary). Imaginary numbers are those that have as a factor the square root of negative one, usually written in shorthand with a lowercase i. One third isn’t infinite either, as it’s only equal to 1/3, which is less than one, which is certainly finite. It’s just a (real) number that doesn’t have a terminating decimal expansion (because 3 is coprime with 2 and 5, and decimal expansions are base-10, which is base-(2*5)).

Despite being both real and finite, you’d never be able to find 1/3 exactly on a number line broken into 10 pieces at each step, but the number would still be there. It’s decimal expansion is 0.333…, the value just a bit to the right of 0.33…33 and to the left of 0.33…34, no matter how many 3s are present. (It’s a third of the way between them, actually, every time.) Most rational numbers (which are just fractions) have infinitely long decimal expansions, and all irrational numbers do. All of them are real and finite, though.

So, 0.999,,, is a real number such that if we go looking for it on the number line, we will know that it is to the right of the 0.9 mark, to the right of the 0.99 mark, to the right of the 0.999 mark, and so on for any number of concatenated 9s. We also know that it cannot be more than 1 because otherwise it would start with 1 before the decimal place. So it is the real number that is bigger than any number that can be written 0.999…9, no matter how many 9s, but no bigger than 1. The only such number is 1.

James Lindsay

Report abuse

Just one comment. Those that have an aptitude for philosophy are not likely to have an aptitude for mathematics; the former is qualitative and the latter is quantitative.

“Nature prefers the language of math.” That’s good to know.

Report abuse

I think you have muddled arithmetic and mathematics here. Logic is the mutual pivot point of mathematics and philosophy, having a rigorous sense of value about a parameter and about a process. This is entirely Wittgenstein’s point about the ambiguous signs of purely metaphysical entities.

Report abuse

@jameslindsay #2

Thanks for replying. It was certainly more than I expected and much appreciated.

I disagree. If you’d “never be able to find it”, it cannot also “be there” (on such a number line).

We also know that 1 is to its right and that it is not equal to 1 “because otherwise it would start with 1 before the decimal place” and it doesn’t.

It is also not exactly equal to 1. If it were, it would be found on the number line.

Report abuse

The only part I like is that Lindsy’s comment is bigger than the post itself.

Report abuse

What is being said here, is that

Natural Philosophyhas matured, as modern science has progressively resolved the imponderable questions of old, leaving many of the earlier attempts a philosophical answers refuted and debunked, while others have been expanded and formulated into precise scientific descriptions of the working of the laws of nature in the universe.Scientific methodology is the best way of establishing correct perceptions and understanding of reality, while mathematics calibrates and quantifies the details.

Report abuse

Well, Thato, math is hard. Explaining math is harder.

Report abuse

Simple answer to PeacePecan. You’re confusing representations of a number with the existence of the number. 1/3 is exactly 0.100… — in base 3.

0.1000.. In base 10 cannot be represented in base 2 exactly. That doesn’t mean the number 0.1 doesn’t exist!!!!!!!

Staying in base 10. We start by defining a number as the positive integers. We call that ℤ⁰. Every number in ℤ⁰ except zero has a multiplicative inverse. For the number 3, we write it’s multiplicative inverse as 1/3. That is 3 * 1/3 = 1. For every number, it’s inverse is unique. Since 1/3 can be written as 0.3333333333….., and 3 * 0.333333333….. is 0.99999999999, by uniqueness, therefore 0.9999999999 …… must be one exactly.

Report abuse

This type of argument does look rather familiar!!!

https://www.richarddawkins.net/2016/02/when-will-the-universe-end-not-for-at-least-2-8-billion-years/#li-comment-199437

(See Alan4discussion #78 of that discussion)

Report abuse

People who are good at science and maths tend to be more logical, weigh up evidence, data handling experience, decent critical thinking skills, good researchers getting the crux of a matter.

Mathematicians are fond of one word in particular- rigor. No grey area, ambiguity and every step started point is explicitly stated.

Lets of useful qualities to have that can be applied from anything from Journalism to gambling.

When I understand what philosophy really is and what has been achieved so far I probably would be able to make a better, more specific comment relating to the piece.

Report abuse

Phil #4, others

Five

Likes,Phil. Good job.“Logic is the mutual pivot point of mathematics and philosophy, having a rigorous sense of value about a parameter and about a process. This is entirely Wittgenstein’s point about the ambiguous signs of purely metaphysical entities.”

I don’t know if I’d phrase it quite that way but I think I agree. How could one not? Logic is essential to honest thought as are “parameters.”

For the umpteenth time, metaphysics is not positive knowledge; it can only be negative. The “parameters” of reason prohibit us from being able to know what any physical object is independently of all minds. Philosophy, that is, critical idealistic philosophy, has

limitativevalue. The phrase: “Pure being cannot be said to be physical” is not dogmatic – not if that conclusion is based on a careful and logical analysis of the nature of the animal mind and its limitations (“parameters” – in my sense).Wittgenstein. I find it all completely incomprehensible, and offensive – to the point that I am tempted to toss my copy of the so-called Investigations in the trash. Why is he famous? Because he is not intelligible, and that is mistaken for genius.

Here are some examples of W’s greatness. These are from his Philosophical Investigations:

“If someone were to advance

thesesin philosophy, it would never be possible to debate them, because everyone would agree with them.”It’s as if this unhappy, obscure man, this darling of the post-modernists, this mind-destroying purveyor of intellectual pollution, who ate spam out of a can, wanted to take his anger out on the world, and did so by confusing our minds. He just said whatever he wanted.

“It is not the business of philosophy to resolve a contradiction by means of a mathematical or logico-mathematical discovery, but to render surveyable the state of mathematics that troubles us – the state of affairs before the contradiction is resolved.”

“Our clear and simple language-games are not preliminary studies for a future regimentation of language – as it were, first approximations, ignoring friction and air resistance. Rather, the language-games stand there as objects of comparison – as a sort of yardstick; not as a preconception to which reality must correspond. (The dogmatism into which we fall so easily in doing philosophy.)”

Report abuse

Dan

And for the umpteenth and one time this is about the looseness of definition of properties of mooted (imagined) entities. This undercuts their value as any “proof” positive

ornegative. It merely suggests….Of W I only ever repeatedly use this argument of his, because it is true. I have no particular interest in defending him though we can go around the post modern jibe one more time. (Lyotard’s ultimate misuse of W is not for W to answer.)

Really Dan, you should work on the campaign team for some political contender. Your ability to throw unrelated ordure is legion.

Report abuse

As Carl Jung said:

He [Schopenhauer] was the first to speak of the suffering of the world, which visibly and glaringly surrounds us, and of confusion, passion, evil — all those things which the [other philosophers] hardly seemed to notice and always tried to resolve into all-embracing harmony […]. Here at last was a philosopher who had the courage to see that all was not for the best in the fundaments of the universe.

[Memories, Dreams, Reflections, Vintage Books, 1961, p. 69]

Now someone please tell me what those particular issues have to do with mathematics which is as useful as it is abstract and removed from anything human.

Report abuse

Phil,

I really don’t know whether W. influenced post modernism or not; I assume he did. Anyway, I am entitled to my honest opinion. You have quite a few opinions yourself. W (and Dickens) have more than enough worshippers, and they’re not going anywhere. A little iconoclasm has its place.

W’s point is true? Prove it.

Not everything is capable of proof. Some things can

onlybe suggested and then rejected or not as a real possibility.Report abuse

Dan #14

Everything, if we are to be certain of any grounding facts in the case. Schopenhauer’s notice of suffering is far from rigorous and hugely subjective. His own probable (and probably unhappy) pederasty leads him to clutch at any old explanation for nature contriving this coupling of old men and young boys. The sperm he observes of young boys is as weak as that of old men and not fit for reproduction. Nature provides another outlet for both in bringing them together, blithely overlooking old homosexuality and young homosexuality as providing a more obvious solution. His view of romantic, consumated love was an uninformed disgrace. He spoke only of what was in his head, and though this was capacious, it was very particular as were its pronouncements.

His offerings could have been so much richer as a philosopher if he had had the solipsistectomy of the scientist.

If, however, he had been a novelist with all the contingency that implies he might have found a greater if more cautious take up of his ideas.

Report abuse

I believe its true. This is why I use it.

My opinion on Dickens, I asserted reflected my differing aspie needs for reliable and pertinent insight into heads. This is not a judgmental opinion on Dickens so much as a comment on my own poor brain and how it affects my aesthetics and desires. You paint it as some sort of offending arrogance.

Lyotard’s use of W’ language games is well documented. W’s creation of these had no intended use for Lyotard’s nonsense.

Bollocks sells better with a little of real value nicked and mixed in.

Report abuse

Dan, re#16

Sorry I answered for “maths and science” not your “maths” which I think a little unfair. Logic is always in need of rigor. Our brains haven’t a single strictly logical circuit and this must be induced in as a “soft” machine of the strictest behaviour.

Logic was taught at my school in maths classes. It also figured in an ordinary maths degree course we were given at university “for free” due to a surplus of mathematicians. Even those in the English department were given lectures in logic. (Somehow the mathematicians seemed to have just multiplied.)

Report abuse

@Phil 16

I read Schopenhauer’s biography years ago. I don’t remember anything about “his own pederasty.” His theory of homosexuality (which I do not remember encountering) may not have any validity, but at least he was writing about it at at a time when almost no one else was. His general views on sexuality, which resemble Freud’s in some ways, stands on its own, and the “undesirability” of procreation, does play a role.

Later in life he had developed, before Freud, an understanding of the connection between sex and death. He never wrote about it, however. Some of his important views on “Romantic Love” (sexuality, reproduction, etc.) entirely support Darwin’s own views. “Schopenhauer used his insight that individual organisms were irrelevant to explain the power of sexual love.” Even Grayling admired S’s “common sense.” I could go on.

I think you are just angry, attacking the man’s state of mind and drawing attention to subjective motives, which are the product of speculation, your own subjectivity. This is often done: “Nietzsche was crazy, therefore… S was miserable, that is why…” Although one’s ideas are often borne out of ill-will, and people do, as we all know, have their axes to grind, it is as effective as it is improper to

a prioriinvoke nefarious or personal motives to a thinker, rather than addressthe ideas themselves.(I did it myself with W. In my case I was speculating merely and expressing disgust. I have a personal distaste for W. but have never tried to repudiate him. I am content to say that and that I find him incomprehensible.)As for S’s observation and conclusion concerning the Suffering of the World, just consider what animals and insects have to go through on a daily basis. —That was a point he often made; the creatures of the earth are continuously preying upon each other, seeking to devour each other. Proof? Watch any nature documentary. That should be enough to inform any reasonable person about the essential antagonism associated with the fundamental nature of existence, if he or she thinks about it long enough.

One may prefer to remark on the beauty of nature (which does exist) and regard nature as a

mere “peep-show.” (S)

What is that quote you have there (16)? Kierkegaard influenced post modernism? Kierkegaard was a religious writer essentially, an austere figure, wrote about faith, the dialectics of faith, was an absolutist, a purist, was critical of “science.” – was a great artist, ironist and humorist too. Very complex figure What’s he doing there? Where’d you dig that one up?

Report abuse

Phil,

There’s been a little discord lately. As a way of making an amends for any insensitivity on my end I offer you this, from

our mutual friendPaul.“Logic is the mutual pivot point of mathematics and philosophy, having a rigorous sense of value about a parameter and about a process. This is entirely Wittgenstein’s point about the ambiguous signs of purely metaphysical entities.”—P.R.Phil is clear if gnomic. Read his statement with more care?

Logic: if a, then b; so, if c then a, then, if c, then b. An absolute. We want to find logic as an absolute basis. Philosophy’s reach for metaphysical base terms fails, however, because without a

“rigorous sense of value about a parameter and about a process”

developed thru use, eg habit, as math has, such terms as Philosophy comes up with are just descriptions of phenomenally complicated chess pieces before they’re actually used. They are untethered by habit.

So to speak.

P

Report abuse

(Peter Boghossian and James Lindsay, article)

This is, despite my upbringing and education in a culture in which mathematics and science were regarded as of peripheral concern in life, the best advice to follow, if one wishes to be knowledgeable and wise about “life, the world and everything.” It is, incidentally, also a call to the very origins of Western philosophy, before Socrates took critical thinking from the cosmic and protoscientific concerns of most of the pre-Socratics and applied it with good effect to ethical questions, and before Plato decided that sense-knowledge was too unreliable and too far beneath the dignity of the soul that contemplates eternally true Forms. Aristotle was more prosaic in his aims than his great teacher and retained the pre-Socratics’ curiosity about physical facts and the need to understand and categorize them. Many of the pre-Socratics were fine mathematicians and astronomers; the earliest named Greek philosopher was Thales, who is best known for predicting an eclipse in 585 BCE.

Before the development of critical thinking, mathematics and scientific enquiry, superstition was the inevitable outcome of our large, unusually complex brains with their evolved mechanisms for pattern-recognition and agency-attribution for rapid decisions about the safety or danger of things encountered, and the evolved respect for authority that enabled dependent children to learn skills and information needed for surviving in the world once they attained adulthood. In ancient Hellenism we see the beginnings of a way of thinking that challenged the superstitions that had been and still were prevalent among humans. Plato’s idealism was an interesting development alongside all the other philosophical and protoscientific developments in ancient Hellenism but, in eschewing empiricism, it represented in fact a return to a mathematicized form of superstition, whereby the real existence of eternally existent, perfect, immutable Forms was asserted without real evidence. This is the basic flaw in any kind of idealism. Mathematics itself, like any form of rationality, can, as Stephen Fry sagely pointed out in this interview, be made into a superstition if it is not accompanied by empiricism. Hence, the history of philosophy is strewn with examples of groundless and useless speculations, however logical, complex, sophisticated, imaginative, emotionally gratifying, entertaining and fascinating they may be. Big brains, like big muscles, like to be exercised and given a good workout, but, until the scientific method (to which empiricism is integral) had been defined, the light that philosophy threw on reality was far from adequate for a systematic understanding of the world and of ourselves.

Report abuse

Hi, Cairsley, (# 21)

I read your comment. Nice prose.

The Platonic Idea is a perception. That is, at least, one interpretation. It is not conceptual knowledge.

“…strewn with examples of groundless and useless speculations, however logical…” Groundless yet logical? “Imaginative, fascinating, complex, sophisticated” yet useless?

This maddening and pervasive anti-philosophy complex that I am encountering everywhere – particularly amongst physicists, atheists (of which I am one) and neuroscientists – is irrational. It is a bias. That’s my honest opinion.

Apprehension of the Platonic Idea, which you regard as groundless and as superstition, has been described this way:

“Raised up by the power of mind, we relinquish the ordinary way of considering things, and cease to follow merely their relations to one another, whose final goal is always the relation to our own will. Thus we no longer consider the where, the when, the why, and the whither in things, but simply and solely the what. Further, we do not let abstract thought, the concepts of reason, take possession of our consciousness, but instead of all this, devote the whole power of our mind to perception, sink ourselves completely therein, and let our whole consciousness be filled by the calm contemplation of the natural object actually present […]”

Report abuse

Dan, as ever, I am poor at detecting potential conflict. When I seek to say things plainly, I seem always to rile you with some kind of seeming value judgment that I didn’t intend. I like saying things square on. When I say nice I mean an agreeable precision, not just pleasant. When I use plastic as an adjective, I mean formable not tacky or cheap. When I say soap opera I mean a mass consumption narrative of many characters published in parts often started without a known conclusion. If I intended a slight, I would include a properly denigrating word, otherwise it is not a slight. You are astonishingly quick to take offence, I cannot keep up. Nor can we lay out the facts of the case and then come to judgment. It may help to know that since teenagerdom the concept of high and low culture has become meaningless to me and most social commentators I know (not pomo…value is still to be discerned). Artists and art consumers couldn’t care less about the concept. It is those self-boostering (denigrating adjective) critics that need the split.

Schopenhauer.

S was a great mind. I have praised his achievements. If he had been a better scientist he would have gone further. He would have worked harder to find alternative and more detailed theories of oppositional colour perception. He would have produced richer alternative accounts for sexuality.

I have no anger here at all. (You give me a much more florid interior life than is warranted.) You chose S. I merely wanted to show that all philosohers (even the best of them) could benefit from a more rigorous (scientific) selection of input materials and a wider embrace of hypotheses (to enhance confidence by elimination.) Scientists would be better scientists if they were more like this. In Schopenhauers day they were rather less scientific than they are now.

Dennett (a favourite and helpful modern philosopher) got his science wrong on Mary the Colour Scientist and got his philosophy wrong as a result. Paul Churchland got it right and his philosophy from the thought experiment now has more chance of leading to something substantial…real.

Scientists, even the greatest, are often outgrown by the accumulation of better evidence and theory. Some philosophy, as the might-bes of its metaphysics collapse like wave functions into empirical reality, sometimes needs to be outgrown also. Philosophers are slower than scientists to recognise this. Philosophy would progress the faster if they better adopted this scientific sensibility.

Report abuse

Dan.

https://en.wikipedia.org/wiki/Postmodern_philosophy#Precursors

PS S was disgraceful (not a good word) on sexuality primarily because his work omitted the female.

Report abuse

@lawrence_j_winkler #9

Actually, I’m rejecting the representation.

This “proof” is circular. It uses it’s own “logic” to “prove” it’s logic. Before I can accept the “proof”, I have to accept the premise that “one third” (or 1/3) and 0.333… are the same thing and that “two thirds” (2/3) and 0.666… are the same thing. There is no “proof” offered for this, other than a

definitional“proof”. (It’s true because we say it’s true.) We might as well skip this “proof” and just use that same definitional proof to “prove” that 0.999… is the same as 1. But what is the (logical)basisfor that definitional proof?Necessity, apparently. It’s “true” because it has to be “true” for everything to work out the way we

knowit should.It’s the

representationsof truth that are uncomfortable.Report abuse

Phil,

Scientists are so clueless about philosophy it’s not even funny; it’s staggering. It works both ways. And to suggest that philosophers need to be better scientists may be true – depending on what problem they are addressing and how they are approaching it – but may not be; was Nietzsche incapable of forming valuable insights about life, growth, decline? Does a man describing or explaining a painting need to be a scientist in order to edify us?

S’s discussion of sexuality is a disgrace? What work are you referring to? There are three chapters in WWR, Vol.2: Life of the Species, The Hereditary Nature of Qualities, and the Metaphysics of Sexual Love. There is, if I recall, an additional essay or two somewhere (I think in

Parerga and paralipomena.I have to look it up.) Where did you get that criticism from, an article, wiki? A lot of opinions out there. He discusses the individual versus the species, says much that Darwin would have agreed with. How does he “omit the female”?Speciesandindividualsinclude the female, don’t they?Report abuse

Dan,

You are obsessed with all that is irrelevant.

Since you seem to have grasped that Wittgenstein et all…have little to say that cannot be dismissed as utter bollocks…read some maths papers and wonder at the empirical proof that will appear before your very eyes!

Mind you, your mind is clearly fundamentally different as mine if you managed to get through “Tess” without fantasising like I did that if you could go back in time and kill one person, that person would be Hardy!

Report abuse

@lawrence_j_winkler #9 – You’re confusing representations of a number with the existence of the number.The representation is irrelevant to the existence of the number! –

As Lawrence points out in the part of his comment you do not quote, numbers can be represented in more than one base, and exist independently of the base used to represent them.

Report abuse

@M27Holts #27

I never said Wittgenstein

et alwas (to use your phrase) “bollocks”; I said Wittgenstein – and I am probably wrong to dismiss him in his entirety.Comment about Hardy disturbing, although I’ve fantasized about striking Wittgenstein.

Thanks.

P.S. Yes, many philosophers have been dogmatic. Dogmatism and groundless assertions should be avoided.

Report abuse

Dan

But all the time the logic of your comments here falls short. Sometimes the scientific mindset brings a cooling distance to seeing exactly the issue.

The question at hand is about better philosophising, so, could Nietzsche have formed more valuable insights if etc….?

“a man describing a painting” is philosophy? A man philosophically analysing a painting, too right!

Too true.

Penrose on consciousness is simply pants. Any number of illogical assertions from scientists need improvement. We’ve had some good discussions in the past here about this. Philosophy departments and science departments are increasingly getting it together.

It tends to be more practical than you might like though….

Report abuse

A man describing a painting is not a philosopher per se. A philosopher describing some facet of human existence is not a scientist per se. A scientist describing the Big Bang or a cell is not a philosopher per se. But each is dealing with philosophical issues, whether they are aware of it or not, or choose to call them that or not.

Report abuse

Hi Dan [#31],

Nicely put.

😉

Report abuse

@alan4discussion #28

If I’m to accept that 0.999… means the nines after the decimal go on infinitely (“an infinite concatenation of nines” – OP) then I cannot accept that it is also equal to 1.

Report abuse

@Everyone (Cairsley, Peace Pecan, Phil, Stephen, et al)

I see that there has been a discussion on this thread about “the number” represented versus the existing number. How can any number not be represented? If it were not represented it couldn’t be counted.

The discussion is on a level that I am not able to understand, as I am an ignoramus when it comes to maths; but I ask you, Peace Pecan, to explain to me how the number 3 can exist and yet not be represented.

Cairsley, this is what happens when we lose sight of epistemology. Mathematics and science can lapse into dogmatism too, as you quite rightly said. Philosophy, critical philosophy,

preventsdogmatism; it does not foster it.Philosophy needs science and needs to be scientific, i.e. precise and non-dogmatic, and science needs philosophy.

Pythagoras, one of the wisest and most humane men who ever existed, was, finally, a dogmatist. He believed that Number was the First Principle. That is like saying that Wind is the First Principle.

Without philosophy to reign science in, as it were, we will go full circle, and the mathematicians will start talking about numbers as things-in-themselves!

Thanks, Stephen.

Report abuse

@PeacePecan

You show considerable misunderstanding of some simple basic stuff of mathematics. Don’t tell us you’re right and the math is wrong, go and take a course, do some study.

The part that flummoxes you is infinite series – and you weren’t the only one, it didn’t get resolved until the time of Newton. (Phil or someone, fact check this one please, I didn’t look it up).

It’s the three dots. It could be the abbreviation “etc”, for et cetera, this time meaning “and so on, and on, and on in the same way, without end”.

Add these fractions (written using base-10 numbers):

9/10 + 9/100 + 9/1000 + …. (here the dot-dot-dot means follow the pattern, 9 over ever increasing powers of 10). The sum approaches the value 1, but never, in any finite series of terms, does it reach 1. It is always less than 1 by a specific amount. The amount, we can write as

1 – (9/10 + 9/100 + 9/1000 ….)

And it’s greater than zero by an amount that depends on how many terms of the series you can be bothered to add:

1/10, 1/100, 1/1000, ….

Now, if we follow this series of numbers, it approaches zero, never quite reaching zero, as you can see. Imagine an infinite series of such numbers, if you can. Mathematicians imagine infinities all the time, they even identify different kinds of infinities, some more useful than others. Usually infinity is a problem, it means you’ve made a mistake. One divided by zero will show an Error on your calculator, because there is no answer.

This is the root of Zeno’s Paradox I believe. The hare that can’t catch the turtle, or the arrow that can’t find it’s target, I’ve seen it re-cast in various guises.

Taking these series of terms, mathematics introduces the term “limit”, the Limit, as the number of terms “approaches infinity” can be found, and is a useful feature. The limit of 0.999… (where the number of 9 digits approaches infinity, is 1.

Don’t argue, don’t disagree because you don’t like it. Correct the rigour of my description, by all means, go study some maths and you may find a better formulation. That’s just how it works, and disagreeing gets you a fail.

It’s not down to what you can and can’t “accept”. Just suck it up, it’s part of the body of knowledge that is Mathematics, and if you still don’t “accept” it, go join the Amish or something because you’re not “accepting” the essential underpinnings of our technology. And get off the internet, since it’s built from knowledge that you cannot “accept”.

Or you could just shrug, and say “it’s not my area of expertise”. That would work.

Report abuse

On numbers, and representations of numbers (for Dan, I think):

There are many ways to represent the same number. 1111 in binary notation is 15 in decimal notation, or 17 in octal notation (base 8), and is the same as 5 times 3 in any notation with base of 6 or more.

We’re so used to seeing “15” that we usually overlook that is just shorthand notation for 10 + 5, or rather for “1 times 10-to-the-power-of one, plus 5 times 10-to-the-power-of zero”.

The same number (the number of fingers on three typical unmutilated hands) can be – and often is – represented by a single character, the letter F in the realm of computers.

The point of all this is that there are many different ways to represent the same number, and some are more concise than others. A very non-concise representation of three-hands-worth-of-fingers could be 14 plus the infinite series that so confused PPecan, denoted by 0.999… , and there might be a situation in which this is a convenient way to represent it, not that I can think of one now, but such is mathematics, adjust your representation (without error!!!) to suit the task in hand.

Report abuse

@dan #3

You seem to be confusing “aptitude for” with “interest in”.

Report abuse

@OHooligan #36

Yes, it was I who asked that question. (Who else?)

You elaborated (very well) on the various ways that a number can be represented.

Now please tell me how an “existing number” is defined (as opposed to a represented number).

I thank you in advance, although I probably won’t get it.

(Now I’m asking you to explain stuff. That’s kind of nice, actually. This is a good site.) 🙂

Report abuse

Dan #31

Completely agreed. Clear thinking is never inappropriate. But

Neither “describing” nor “dealing with” are synonymous with analysing. Generating philosophy, creating new material, is quite distinct from “dealing with it”.

If you want to argue philosophy and science etc., have their roots in everyday thought, this is true but trivial, nor is it too helpful for analysing, understanding and mastering the problematically distinct disciplines.

Report abuse

Dan

You might have been (dis)pleased to hear Marcus du Sotoy on Jim Al Kahlili’s program “The Life Scientific” declare exactly that. He believes that in some sense mathematics existed as a defining property of the universe before any of its expressions….

https://en.wikipedia.org/wiki/Marcus_du_Sautoy

Atheist, actor, playwright, musician, and the current Simonyi Professor for the Public Understanding of Science… whats not to like?

His “Music of the Primes” was a great book.

Does he need philosophical help with his assertion? Almost certainly, but he is not dogmatic in his “first principle” claim. He merely has a suspicion that it is so… Such suspicions so offered are not the problem. It is claims of certainty.

Report abuse

The argument from base three representations given before is the clearest here.

(All numbers are base three unless they are spelled out when they are base ten.)

In base three we count (to six) 0, 1, 2, 10, 11, 12, 20…

One over (divided by) three in base three is 0.1. If we add it to itself twice we get 0.2 and then 1.0

Report abuse

@Phil

40

Interesting.

39

“Neither ‘describing’ nor ‘dealing with’ are synonymous with analysing.”

No, I suppose they’re not. I don’t want to sound like a pluralist a la W, but there are many modes of philosophic expression. “Clear” thinking is not required. (Kierkegaard was an ironist and employed an indirect, oblique mode of communication.) Nor is consistency. (Nietzsche contradicted himself all the time.)

“… have their roots in everyday thought…” Are you for that or against it? Wasn’t clear.

“The book deals with the problems of philosophy, and shows, I believe, that the reason why these problems are posed is that the logic of our language is misunderstood. The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence.” -Tractatus

(I can’t help feeling that W had some kind of weird agenda. He was a true anti-philosopher. Perhaps he was motivated by revenge, felt slighted. God knows.)

Report abuse

O’Hooligan,

Yep, your are right.

The assymptote for the curve created by the process of endlessly adding another nine to the end of the decimal fraction 0.9 and its subsequent sum , mathematicians know to be one. We are muddled by the idea of an endless process. There is of course no such process. Since Newton and Leibnitz the term “in the limit” has a clear mathematical meaning.

This approach may not be as open handed as the simplicity of a base three description, but it is a more usefully generalised argument.

Report abuse

For those who cant be bothered googling:

http://www.slate.com/articles/health_and_science/science/2014/03/zeno_s_paradox_how_to_explain_the_solution_to_achilles_and_the_tortoise.html

The whole .999…. = 1 puzzle is explained there quite clearly. Among other places, but that’s the first easily readable one I found

Report abuse

Dan

Anti-philosopher…hm…this may be the distinctive qualitiy of a modern scientist as opposed to the natural philosopher of old. The modern scientist creates increased net certainty by disproving hypotheses. This has an analogue in a gardener pruning back to encourage a better growth. It looks destructive but ultimately is quite the reverse.

Its just a fact…nothing to approve or disaprove, just accept. But the plant grows a long way from the roots. More to the point we then get to prune it to suit our needs….

My Thursday night “liquid philosophy” sessions are also philosophy in this broad definition you are happy with. But there really is an attempt to achieve a greater reliability to musings by rooting them now not in unkempt thought alone but in observable reality.

Report abuse

Your skill at asking weird questions is impressive.

You want to hear that numbers only exist in the mind, don’t you? Since they are abstractions used to enable mental trickery to work out answers to real-world quantitative questions without needing physical tokens to count, such as beans, or apples.

I defer to Marcus du Sotoy, he knows more about this than I do, if his qualifications are to be trusted. That’s not an argument from authority, I just think he’s a better explainer.

Report abuse

On math, for true mental trickery, wait til you find out about Imaginary Numbers.

Impossible, but this notion provides another set of mathematical tools for solving real-world problems. At the end of a calculation involving both real and imaginary numbers, the imaginary part is discarded, after all, in the end, it was Imaginary. And yet a real part remains, an answer that we know of no other way to achieve except by means of these impossible Imaginary critters.

Its a set of tools, nothing more. An axe is a tool that is useful because wood is the way it is. Math is useful because the universe is the way it is.

Report abuse

The representation is irrelevant to the existence of the number.If the recurring decimal places of 9s go on to infinity, – put simply, that means that there is an infinitely small (ie. non-existent) difference from one!

It also means that this is the most accurate way of describing arithmetic using thirds in base ten. –

Looking at base three etc. illustrates this point.

As I pointed out earlier, you seem to be asserting your misunderstanding, as you did in this earlier discussion:-

https://www.richarddawkins.net/2016/02/when-will-the-universe-end-not-for-at-least-2-8-billion-years/#li-comment-199437

Report abuse

OHool #46

I would say

Quantity and the representation of quantity are obviously distinct.

Report abuse

@PeacePecan

In all of this, there are multiple ways to look at the numbers involved. Mathematicians define them in terms of sets and relationships. Primary school 2+2=4 skips for simplicity’s sake what we mean by “2” and “4” and also what we mean by “+” and “=” … we think we intuitively know what we mean with regard to simple addition and subtraction, dealing as it does with counting and counting-in-reverse, but such simple techniques hit difficulties with the concept of negatives.

Putting numbers on a set-theoretic basis goes something like this. (I’m leaving a lot out, and hope you just connect the dots yourself or google for further information).

There exists an empty set {}. This is an axiom. For convenience, we can label this “0”.

Sets can contain other sets (another axiom), so we immediately have another at our disposal, namely {{}} or (for convenience) {0}.

The set containing the empty set is not the empty set, and for convenience we can label this “1”.

The set {0,1} can be labelled “2”. The set {0,1,2} can be labelled “3”. And so on.

Obviously we end up with a bunch of sets which we have chosen to label 0,1,2,3,4,… which look suspiciously like the natural numbers. But remember they’re sets.

For any two such sets x and y, we can say that x lt y if x is a member of y. This relation lt which looks a lot like the “less than” sign in arithmetic behaves in the same way. Either x lt y or y lt x or x=y. And x lt y, y lt z implies x lt z etc.

We can define addition on the natural numbers, but not subtraction – because “2 – 3” would be undefined.

Anyways, if we look at ordered pairs of natural numbers such as (0,1) and (3,2) we can define an equivalence relation on them and up with something looking suspiciously like the integers. (An equivalence relation ~ is a relationship that is reflexive so that x~x (for any x), symmetric so that x~y implies y~x, and transitive so that x~y and y~z implies x~z.)

An example, all pairs of natural numbers (0,0) and (1,1) and (2,2) etc. are equivalent to each other, and are “like” the integer zero (0). Likewise, all pairs of natural numbers (0,1) and (1,2) and (2,3) etc. are equivalent to each other, and are like the integer one (1). And the pairs of natural numbers (1,0) and (2,1) etc. are like the integer minus one (-1).

We can treat a single equivalence class (all equivalent ordered pairs of natural numbers) as a single integer. And all possible equivalence classes of natural numbers as the set of all possible integers.

Now we can get addition and subtraction and multiplication defined on the integers. But not division, as “2 / 3” would be undefined.

Having defined the integers, we can get the rational numbers in a similar fashion.

The ordered pair of integers (1,4) can be identified with the rational number 1/4, as can of course ordered pairs of integers (2,8) and (3,12) etc. — an entire equivalence class. Rational numbers may be considered as the set of all equivalence classes of ordered pairs of integers, in the same way that integers may be considered as the set of all equivalence classes of ordered pairs of natural numbers.

We can get addition, subtraction, multiplication and division defined on the rationals, but not the square-root function, as “sqr 2” would be undefined.

Moving up to the concept of a series.

The series 0.9, 0.99, 0.999, 0.9999 … converges on 1. It’s not the only series to do so, it’s just one of an entire equivalence class of them. (The canonical series would be 1,1,1,1,…)

The real numbers may be considered as the set of all equivalence classes of series of rationals.

Report abuse

@alan4discussion #48

Despite the exclamation point, infinitely small is not the same as non-existent (and never will be). A difference is a difference no matter how small. If you insist on telling me this is not true, then we will never agree.

(Yes, you do seem to relish making this statement.)

This is a discussion forum. I read the articles that interest me and when I disagree with their assertions I assert what I believe to be true, and/or ask for help understanding the stuff I don’t understand, just as many others do. (I find little value in posting agreement, though I have done so at times.) When presented with good reason to change my belief, I do.

Report abuse

@MadEnglishman #50

The series 0.9, 0.99, 0.999, 0.9999 …

convergeson 1.Yes.

Report abuse

Clearly you do not understand the concept of “infinitely small and continuing to diminish in infinite succession” to zero.

Not really. It is more of an expression of frustration, when others and myself, have presented good reasons for you to change your “beliefs” without effecting any progress.

Report abuse

I’ll give this another shot with a concrete experiment.

Take 3 precisely engineered 1 metre rulers – calibrated in metres and centimetres on one side.

Turn 2 of them of them over and on the blank side mark them off in thirds. 1 third of a metre, 2 thirds of a metre 1 metre.

Precisely cut off a one third of a metre from one at the one third mark, and discard the other piece which is now shortened by the width of the cut.

Precisely cut the second ruler at the 2 thirds mark and discard the spare cut off end which is shortened by the width of the cut.

The first will be precisely one third of a metre long as is indicated on the base 3 side, and 0.33recurring metres long on the base ten side.

The second will be two thirds of a metre long as shown on the base 3 side and 0.66 recurring metres on the base ten side.

Both of these cuts will be at a precise material point, regardless of which numerical base is used.

Place these two pieces end to end on top of the third uncut ruler.

They can be checked and shown to be the same length, but the numbers on the pieces (0.33recurring m + 0’66recurring m) will add up to 0.99 recurring m., while the identical uncut ruler will show 1.0m for the identical length!

In the material world, points exist on number lines, regardless of if a figure is present at that point of measurement in a particular number base.

Report abuse

@PeacePecan

Your “beliefs” regarding mathematics are irrelevant. You’ve had plenty of coaching on this forum. The awkward concept for you seems to be the use of infinity in mathematics, a concept that doesn’t really have a real-world equivalent in a finite universe (if it is finite), but, by giving it a name and a symbol it becomes part of the toolkit that is mathematics.

“The series 0.999… converges on 1” you affirmed. That’s it, that’s all, the Limit, as the series “approaches infinity” is 1. The 3 dots meant the whole series, infinitely long. It’s not incorrect to say that 0.999… equals 1, as the difference between them is zero.

See, it wasn’t that hard. As my math teacher often put it: “Now, lads, let’s do a little trick”, and he’d introduce a new concept, a new tool, to attack a problem he’d just set out, that we were as yet unequipped to solve. Why this new trick? Because it works, no other reason. Each trick was firmly rooted in what had gone before, each trick was explained (“nothing up my sleeve”), and put to use thereafter.

Thus I always saw math as a bag of tricks, a conjuring set, or a toolkit, one that could be mastered to deliver correct results, and when I reached Applied Mathematics, useful results.

Report abuse

@alan4discussion #53

It’s not “to zero”, as if it actually reaches zero. It approaches zero, but

neverreaches zero. That’s what I understand.Report abuse

@ohooligan #55

The difference between them is not zero. It approaches zero, but

neveractually reaches it. (That was what I “affirmed”.)Report abuse

@peacepecan #57

You continue to mistake the meaning of the three dots. 0.999… is a cumbersome representation, it is an infinite sequence of 9’s after the decimal point. And at that infinity, the value of the expression is 1, not nearly one, but exactly one.

It never “actually” (for any finite sequence of 9’s) reaches one, correct. But for the infinite sequence denoted by the abbreviation “…” it does. Otherwise Achilles never catches up with the tortoise in Zeno’s Paradox.

The mathematical “trick” is in discovering that a sum with an infinite number of ever decreasing terms CAN AND DOES add up to a finite value. And a very useful trick it turned out to be. Just because you can

neverwrite out an infinite sequence, doesn’t mean you can’t do the sum, and get the only possible answer, which in this case is 1.I’m feeling like a remedial math teacher trying to explain this to you. I’m not, I’m just someone with a bachelor’s degree in mathematics, and I’ve never taught since graduating, so my explanatory skills may be a bit rough. Anyway this is mid-high-school level stuff, as far as I can recall, nothing very advanced.

Please do your homework. Apply the rules and you’ll get the right answer. It’s not a topic for debate, we’re not going to Teach the Controversy, there isn’t one, just understanding and misunderstanding. I hope you’re now able to move on from the latter to the former. Peace.

Report abuse

PeacePecan #56

It approaches? This is not a process that happens in time. Even if it were your “never gets there” is a random selection of time and not at infinite time.

How do we write a third as a fractional decimal?

What is three times that?

(Answers, point three recurring, point nine recurring,)

Report abuse

Phil, OHooligan

In the real word both matter and space are infinitely divisible. I doubt if that is relevant, or that modern physicists and mathematicians would be likely to agree. Just putting that out there.

One is the loneliest number. Discuss? And someone called me a “big fat zero” the other night! Phil, was that an insult?

Report abuse

Dan #60

I don’t think it is relevant. Mathematicians will agree and happily even play now with infinities and infinitessamals of different classes each clearly valued as bigger or smaller than the other. But physicists know your statement is incorrect and that stuff is quantised and panic at the size of the planck length 1.616 e-35m. This is the grain size of a granular space. An electron, however, is 5.6e-15m over 300,000,000,000,000,000,000 times bigger than this.

Probably rhyming slang for hero.

Report abuse

[Sorry I inadvertently annoyed you on the other thread, Phil.]

Report abuse

Clearly you do not understand the concept of “infinitely small and continuing to diminish in infinite succession” to zero.As Phil points out @#59, your claim of “never actually reaches zero”, inappropriately assumes a time scale.

The “approach” is instantaneous and takes zero time, reducing the gap an infinite number of times in zero time.

That’s what what you misunderstand and erroneously assert, on the basis of semantic misconceptions rather than mathematical understanding.

I thought my simple concrete example @#54, should have made this clear.

The comparison of rulers was simultaneous.

Report abuse

In Brief:

1/3 + 2/3 = 3/3 = 1

1/3 = 0.333… (the dots representing the

infinitesequence of 3s)2/3 = 0.666… (likewise)

— ——- (add them up, normal arithmetic, no carry required as 3+6 = 9 in each position)

3/3 = 0.999…

since 3/3 = 1, we have 0.999… = 1

QED.

Not so hard after all.

Report abuse

@phil-rimmer #59

I hope you understand that I’m not the one who introduced the word “approaches” in this discussion. Is it unreasonable to interpret this word as meaning “gets ever closer but never actually arrives”? Thank you for helping me understand that time is not involved, despite all the words used that suggest it is (e.g. infinity, expansion, converge, approach, continuing).

Report abuse

@danielr-2 #60

I think infinity must be the loneliest number: it is always “approached” and “converged” upon, but never reached (unless you’re a mathematician – and who wants to hang out with one of them?).

Report abuse

I hope that was sarcasm and not sour grapes, my peaceful nut. Mathematicians are people too, who do all kinds of stuff that might surprise you. One I know plays bass in a hard rock band. But then again, who wants to hang out with bass players?

Report abuse

@ohooligan #67

Yes, it was. (And not very good sarcasm, at that.)

Report abuse