numberphile : do numbers EXIST
a) You missed a shade between Nominalism and Platonism, namely Aristotelianism. A Platonist would argue that any set of three objects has its threeness directly derived from Holy Trinity (if he is a Christian).
A true Nominalist would argue that any set of three objects is indeed a set of three objects if you call it that, but that the threeness of three sisters and the threeness of three dots on dice are not just physically separate, but the concept is only the same to us.
Whereas an Aristotelian would argue that any set of three things shares a basic threeness with any other set of three things.
A threeness which to a Christian Aristotelian is an image of Holy Trinity, but not necessarily a participation.
b) you do the mistake of calling "sqr root of two"* a number when it is a proportion and "sqr root of minus one" a number when it is an algebraic fiction.
Just because "sqr root of minus one" is a fiction does not mean "three" is a fiction.
*Sqr root of two is diagonal of a square compared to one quarter of circumference. And π is circumference of circle compared to its "diagonal" or diameter. These proportions are always the same, they are quite as reliable concepts as three, but they are geometrical and not arithmetical. They do not count separate quantities but measure continuous ones. Therefore the only exact value of them is precisely those names. Try to use it in arithmetic operations rather than geometric demonstrations, you need a value and that must be numeralised, that numeralisation is a fiction (because numeralising what is in itself not a number) and it results in an approximation. And in the sphere of physics and technology, where these claculations apply, approximation works.
numberphile : root 2
Ha, first of all, Euclid and Aristotle were no Pythagoreans.
Second of all, Pythagoreans were pretty much a freemasonry.
But, mathematically, even if ratios are number to number ratios, they are not numbers.
A triangle 3-4-5 does not have three separate items on one side and so on, but here numbers are used as comparisons. And obviously comparisons can be divided into the continuum. Some are non-numeric ratios, like pi or sqrt of two.
That is pi or sqrt of two rationally defined.
Now, your problem is whether you can rationally define complex numbers.
Can you rationally define zero or minus one as numbers, even?
Of course you can "redefine number" as some try to redefine marriage.
But you can't make two men beget children and you can't make zero or minus one rational answers to rationally asked question "how many".
Giordano Bruno was burned for saying that each universe (what he called universes is what you would call solar systems, supposing each star to have exoplanets) had a soul and that God was the soul of our solar system, or in his words, of our universe, but another universe had another soul and therefore another God.
Both pantheism and polytheism in a baptised Christian.
And of course he did NOT prove any of these. Any more than zero has been proven a number.
- Pi and sqrt(2) are non-numeric ratios? What definitions are you using for number, ratios, and non-numeric? I see that in another post you seem to say that being able to answer the question "how many" defines it as a number. Alright. Let's define a unit length. We can copy this length and thereby ask "how many unit lengths is this segment?" Next, construct a square with a side of 1 unit length. Connect any of the two diagonals. How many unit lengths is this diagonal? sqrt(2), no?
- Hans-Georg Lundahl:
- Number: narrow definition "many" (as opposed to one). Broad definition: "one or many".
Ratio: relation of two quantities (discreet as numbers or continuous as lengths, weights, etc.) such that relation remains the same if both quantites are augmented or diminished by same *ratio*.
As such a ratio may be numeric and also fall between the numeric ratios.
"Adding" centimeter to centimeter may accidentally say "how many centimeters", but really "how big in ratio to the centimeter". Not nec. a #.
- As for pi, construct a circle with a radius of 1/2 unit length. How many unit lengths is the circumference? pi. I've outlined how to construct pi and sqrt(2) in such a way that answers your question "how many". So, why aren't pi and sqrt(2) numbers? Of course, I've assumed that we would agree on many things. If you disagree with anything I did, feel free to post a rebuttal.
- Hans-Georg Lundahl:
- Pi is not a number. The question "how many unit lengths is the circumference" is a malformed question. Pi would be the only correct response, but it is only by making it the repsonse to this malformed question that you make it a "number" at all.
The proper question with measurements is "what is the ratio" (either between two concerned measurements or between one concerned and one standard unit length).
And the correct answer to that correct question is: "pi is the ratio, and it is irrational."
- Wait a sec... if the sqrt of 2 is a irracional number how can it be represented by a fraction longEdge/shortEdge?
- Hans-Georg Lundahl:
- Because long edge and short edge are not necessarily related in a numeric way. On A4 format they are, ideally, not.
- Hans i see some of your comments and you're being kind of ridiculous. The current definition of number is not singularly defined by you. Why are you arguing for your incredibly archaic definition of it. Meanings and definitions change over time as new information is gained and changes.
All ratios are numbers, Pi is a number. Irrational NUMBERs are numbers. It is so much a number in fact it's part of a set called 'Real Numbers'.
- Hans-Georg Lundahl
- I would call "real numbers" unreal numbers. They are however real ratios.
Excepting of course when they are in fact natural numbers.
"Meanings and definitions change over time as new information is gained and changes."
If that were so it would not be a gain of information. It would be a shift of wording in existing information. ONLY if definitions rest the same can informations duly augment.
I am not trying to "define the current" meaning, but to adhere to the Classic one (set by Aristotle).
- I see what you're saying about ratios and measures, but I'm not buying that pi isn't a number. If i did buy into your ideas, then I would also say that 1 isn't a number. 2,3,4,5,6,7,8... and so on would also not be numbers. They are just ratios to the unit length. If I have 3 watermelons, then it's just a ratio of 3:1 with my unit watermelon. Anyways, this doesn't really seem like a matter any of us will be able to settle decisively. It comes down to the arbitrary notion of number.
- Hans-Georg Lundahl
- When 2 is twice as long as the unit length, it is not a number but a length. It is in a numeric ratio to the unit length. Between numeric ratios to it (like 1:1, 2:1, 3:2) there are also non numeric ratios to it, like sqr rt of two or pi.
When 2 is twice as many as one single, then it is truly a number.
In "2 oranges" 2 is a number, in "2 cm" 2 is a numeric ratio.
- Lauri Markkula:
- So having children makes you married...?
- Hans-Georg Lundahl:
- Not the having them per se, but the purpose in advance of both begetting and raising them together in lifeling fidelity, expressed before one's relevant communities (above all Church, for Christians) by an act called wedding which includes a mutual promise.
- Lauri Markkula:
- What about the people who don't get children? What if either the man or the woman is sterile? What if the woman gets married after menopause? You can't get sterile men and barren women to beget children. Should we not ban marriage between them?
- Hans-Georg Lundahl:
- As to people who voluntarily sterilise themselves, the Catholic Church already considers that a mortal sin, and marriages contracted after such an illdeed are null and void, due to lack of intent.
As to women after menopause, the intent need not be lacking if for instance one were prepared to accept children begotten by miracle (Sarah, Elisabeth, St Anne the grandmother of Christ) or after reactivating ovaries by hormonal therapy, which sometimes has that effect.
numberphile : problems with zero
What about my radical solution: zero is NOT a number. It is perfectly good as a "numerical relative", like in addition/subtraction +/- zero means "same as previous, no difference", just as */: 1 in multiplication/division.
It is also perfectly good as a value, geometric or similar (most famously thermometers, perhaps) where "zero" as a value is very far from any real zero of whatever it is a value of.
But it is neither one, nor many, hence no number. Call it a number, you get these problems.
[added one day later after getting no answer:]
All here numberline fundies (and even Gaussians)?
No Roman Numerals fans?
Appendix I, square root of two:
In arithmetic there is none such. Reason, square number and square root number are interrelated. Just as only even numbers are double numbers that have half numbers, so only square numbers can have root numbers.
In geometry there is one. As two squares can be any ratio to each other, they can be the ratio of 2:1. In that case the sides are in a ratio of square root of two to one. And as any paper lover knows, there is paper where the rectangular sides of same paper are that ratio.
Now, square root of two is often given as 1.414... and that obviously only applies to geometry, since arithmetic offers no inbetweens between 1 and 2.
It is also as obviously an approximation, because it is an irrational geometric ratio. And it is irrational because in arithmetic 2 is no square number.
But since it is a question of geometry, it is purely conventional, and apt to confuse it with arithmetic, to give that approximation in decimal fractions. If we use feet, the fractions are given in duodecimals.
One square has side one foot, and obviously the surface one square foot. What is the side of a square the surface two square feet?
The rough approximation offered by using whole inches would be 17 inches or one foot and five inches. It gives a square of 289 square inches. One square inch too many, since two square feet are 288 square inches.
A little finer, use lines as well, twelve lines to the inch just as you have twelve inches to the foot.
(1 ft, 4 in, 11 li)2 = 203 li*203 li = 41,209 li2 which is nearly two square inches too little.
If England had had the older French system, there would have been also 12 points to the line.
1 ft, 4 in, 11 li, 7 pt = 2443 pt is the nether approximation.
2444 pt is obviously the upper one.
If one square foot = 2,985,984 pt2, then two ft2 = 5,971,968 pt2.
Square the two approximations, you will get 5,968,249 pt2 for the nether and 5,973,136 pt2 for the upper one.
And as square root of two is irrational, there is no exact number of subdivisions of the foot that can be exactly the answer. But that does not really matter practically, since the points are closer together than one milimeter. The approximation is more exact than the thickness of an ordinary pencil./HGL
Appendix II, "irrational numbers" are non-numeric ratios between numeric ones (table, two examples)
|4:1||> π >||3:1|
|32:10||> π >||31:10|
|315:100||> π >||314:100|
|3142:1000||> π >||3141:1000|
|2:1||> sqrt (2) >||1:1|
|15:10||> sqrt (2) >||14:10|
|142:100||> sqrt (2) >||141:100|
|1415:1000||> sqrt (2) >||1414:1000|
Appendix III, proof sine, cosine, tangent are not numeric per se
You recall the Amerindian chief Sohcahtoa?
OK, there are angles for which all of these are numeric ratios, like the narrowest angle of the Egyptian triangle:
SOH, Sine = Opposite/Hypothenuse = in this case 3:5
CAH, Cosine = Adjacent/Hypothenuse = in this case 4:5
TOA, Tangent = Opposite/Adjacent = in this case 3:4
And now there are some math wizzes who will know what I do not know, how great this angle is in degrees.
But as obviously there are triangles with clearly non-numeric ratios. Sine or cosine of 45° is inverse ratio to sqrt (2), since it is a side size (simplest form) 1 to another side which to it has the ratio known as sqrt (2). And 30° will have a sine 1/2 but a cosine or a tangent involving sqrt (3). And these, as previously said, have no reality in arithmetic, properly speaking, in the science of odd and even, of prime and compound, of triangular or square or pyramidic or cubic numbers. They belong to the other branch of pure mathematics, known as geometry.
Added app. II and III on Ascension Day, 9-V-2013./HGL
... on Mathematics and Semantics