For instance, pi = 3.141592... can be defined as the ratio between the length and the diameter of a circle. Any integer can be defined by a finite sequence of figures, and so on.
Dear Steven,
Every mathematical paper contains always some definitions and notations introduced by the corresponding author. In general, these definitions must be only considered in the paper context. Unfortunately, there are a limited symbol set to be used, and this fact oblige us to term different objects by the same symbols. This inconvenient does not matter whenever the author takes care of defining them.
The great french mathematician Henri Poincaré says: "Mathematics is the art of denoting different things by the same name". Of course, he was thinking in equivalence classes and analogies. Analogies are also particular cases of equivalences.
The father of normed spaces, Banach, wrote the following:
A mathematician is a person who can find analogies between theorems;
a better mathematician is one who can see analogies between proofs
and the best mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one who can
see analogies between analogies.
(Stefan Banach 1892 - 1945)
Best regards.
Juan Esteban
P.S. I have sent you my paper about Cantor's theorem via e-mail.
What do you think about this equivalence Stefan = Steven = Esteban?
It is not a matter of suppositions. Axioms define the generic concept of real number, but not every particular case. For instance, the following are particular cases: The number pi is defined as the ratio between the diameter and the radius of a circle; the square root of 2 can be defined as the hypotenuse of an isosceles right triangle, and so on. Can you prove that there is a finite definition for every real number?
Sir,God create the numbers and rest are all men, precisely when man saw that ratio between the diameters and the radius of the circle is constant and it's value,which is an irrational number nearly 3.14, then they define it as a pi for sake of simplicity,as because it is not possible to use the exact value in our calculation,similar conclusion can also be drawn for root of 2, can anybody dare to use the exact value of these numbers? Even if somebody still search for a definition of real number then I personally suggest him please construct a equation(algebraic or trigonometric) whose one of the root is that number.......
It is not a matter of faith, but proof. Opinions are not worth in science. If you have any proof for your proclaim, please give me an algebraic or trigonometric equation to define the real number the binary figures of which are 0.c_1c_2....
and they are obtained at random by tossing a coin, as follows. If in the n step the result is F then c_n = 1, otherwise is c_n = 0.
Sir forgive me I have no idea of binary numbers , what is its digitally
Hey RSF It seems me that your English is poor, please don't mind ... what you wanted to say about this topic?
Have you understood the question?
The question is not whether the real number concept accepts a finite definition. The question is if, given any real number R, you can define it by means of a finite set of properties. For instance you can define √2 as the hypothenuse of a right triangle.
Can you find a finite definition for the irrational number 0.101001000100001000001...? Of course you can, because the mantissa satisfies a predictable pattern. What about an irrational number the mantissa of which does not satisfy any predictable pattern? For instance when its figures are chosen at random.
Dear H. Guettinger,
Again I must ask you: Have you understand the question? I am asking for a "finite" definition.
By chance the nested interval method is finite or it consists of infinite nested intervals?
By chance a Cauchy sequence is finite or consistis of an endless string of numbers?
are Dedekind cuts finite methods?
As far I I know, an endless construction can be defined finitely, provided that it satisfy some predictable pattern.
The answer is NO. Chaitin's constant is an example
http://en.wikipedia.org/wiki/Chaitin's_constant
I think you need to think a little more about what you mean when you say 'finite definition'. When you say that pi is the ratio of the circumference to the diameter, this is not exaclty a mathematical definition but merely a physical description. This does not immediately imply that the number pi can be mathematically defined using a finite set of properties. For example, it would be more sound to say that route 2 is the real number satisfying the equation
x^2=2
In any case the answer is indeed no.
I think you mean geometrical instead of physical.
Defining pi as the ratio of the circumference to the diameter is a definition stated more than thousand years before Galileo or Newton. In the Ancient Greece, Archimedes of Syracuse gave an approach to pi, namely 22/7, and defined it as the ratio of the circumference to diameter. For instance, see squaring the circle problem in http://en.wikipedia.org/wiki/Squaring_the_circle
which is a classical one. In addition, Archimedes stated an endless method to compute pi that according to the great mathematician Blaschke was the first example of integral calculus.
Recall that until the introduction of algebra by Muhammad Ibn Mussa Al-khwarizmi in the IX century, math were only geometry, and calculus were done by means of rule and compass. In any case, it does not matter how you define a given real number, my only condition is finiteness. Burnik's answer is fine and very useful to my aim; since it is not an opinion, but he refers a proof.
I'm not disputing that the definition of pi you give is the correct description, just on your assumption of finiteness. You say that this is the condition you are interested in. Certainly your definition of pi is described in finitely many words, but a finite verbal description is not necessarily equal to a finite mathematical definition. That is my only point.
I cannot see any relation between my question and your suggestion. My question is not about sets. It is not a question about cardinality. The topic proposed is about real numbers and not about real number sets. See this example.
Let r_1 and r_2 be two real numbers and 0.c_1c_2...c_n-- and 0.d_2d_2...d_n... the decimal expressions of which respectively. Suppose that both figure sequences, that is to say, the c_n and the d_n are chosen at random. The set {r_1,r_2} is finite, therefore it is a countable set. Nevertheless, according to the answer by Burnik, neither r_1 nor r_2 can be defined or determined by a finite set of statements. As I have said, the definitive answer to my question is that exposed by Burnik. I have found out other complex solutions, but my interest is to know some different one as that exposed by Burnik.
In addition, notice that there is a fuzzy region between a formal language and any colloquial one. For instance, if Pi is defined as the ratio of the circumference to the diameter, then it is absolutely equivalent to the formal expression
π = L/d
where L stand for the circumference length and d denotes the diameter. Algorithms can be also expressed in any programming language and the algorithm notion includes, as a particular case, every finite and efficient procedure to determine any mathematical object.
It is not a matter of words, but statements, that is to say, those sentences that can be either true or false. Notice that a word meaning is always a convention and you can invent a language every word of which contains infinite many letters even infinite meanings. Logic cannot depend on conventions. We are dealing with definitions instead of descriptions.
It is the third time that I must say "the correct answer is the Burnik's one".
If you see some mistake in the Chaltin's constant, see http://en.wikipedia.org/wiki/Chaitin's_constant,
your are welcome to let us know it. If you cannot see a mistake in the Burnik's answer, then the game is over.
In any case, thank you.
A definition is a finite set of statements determining an object. A statement is any sentence that can be true or false.
For instance, you can define the number pi through any of the following sentences.
Pi is the ratio of the circumference and the diameter.
π = C/d
where C is the circumference length and d its diameter.
Both definitions above are finite, since each of which consists of a single sentence.
The following one
π = lim S
where S = 3.0, 3.1, 3.14, 3.141, 3.1415 ......
is infinite, since the sequence S contains infinite many elements, besides, it does not satisfy any predictable pattern to determine any of them.
The number 1.01001000100001000001...... is irrational, however this definition is finite, because the infinite sequence of figures satisfies a predictable pattern that allows to be defined in the finite expression above.
Just one detail, the value of π depends on the metric we use.
The most popular metric for expressing this value is the Euclidean metric which yields π = 3.14159....
http://mathworld.wolfram.com/EuclideanMetric.html
No, you cannot. Despite what has been discussed , this has a direct relation to set theory and more specifically and computability theory (also known as recursion theory). In a nutshell, there are countable many finite "descriptions" (formalized as algorithms) while the reals are uncountable. So there are numbers that are not computable, meaning that you cannot know which numbers those are, there are linked to no property, i.e. there is no way to write those number down descriptively.
See also: http://en.wikipedia.org/wiki/Computable_number
I am about to publish a paper, accepted by International Journal of Open Problems in Computer Science and Mathematics, and in this paper I show that if all real numbers could be determined by any finite process, then the well-known Cantor's theorem would be false. As a consequence there are real numbers that cannot be neither defined nor computed or determined by any finite procedure.
Here you can see the abstract:
http://www.researchgate.net/profile/Juan-Esteban_Palomar_Tarancon/
I stated the question in order to see whether somebody knows any concrete and noticeable example. The Chaitins constant referred by Konrad Burnik is a good example for me. In any case, when I have proposed the question my paper was already accepted by the peer-refreed journal IJOPCM.
P.S.
A simplified version without proofs is available in my researchGate page with the link above.
Utility is not in the scope of pure maths. Thus, I cannot figure out how is possible your conclusion after reading my post-
Of course,
The lack of math symbols compels us to behave as if we were half-mute men.
Dear Hanspeter
The initial question seems to be modified, because several contributors have deviate it towards other ones that are equivalent. This is why I have not thought necessary to re-state the initial question.
For my purposes, it does not matter to substitute my initial question by another one being equivalent.
For instance, the following statements are equivalent to my topic.
1) Does there exist some finite definition for every real number?
2) Does there exist some finite method to determine every real number?
3) Does there exist a finite method to define every infinite set of integers.
The last statement seems that does not fit in this topic. Nevertheless, notice that 3) is equivalent to both 1) and 2).
To see this fact, consider a infinite set of positive integers E = {a_1, a_2, a_3…}. Now, write a real number lying in the unit interval 0.r_1r_2r_3… choosing its figures as follows: if 1 belong to E then r_1=1 otherwise r_1 = 0; if 2 belongs to E then r_2=1, otherwise r_2=0 and so on. As you can see, the problem of defining an infinite integer set is equivalent to the problem of defining a real number lying in [0,1].
This is why whenever the contributors are handling equivalent statements, I have nothing to disagree with.
are you considering "non-terminating" as a synonym of "finite?
If this is not the case, I cannot understand you, otherwise your claim is even more puzzling.
I may not have reviewed the comments sufficiently, but did anyone here define exactly what they are referring to when they say "real number" or "infinite?" Without it much of the discussion here is ungrounded.
In addition, no one seemed to challenge the assertion that all statements are "true" or "false." I assume this was meant to refer to statements that include equality. However, in mathematics (and not even in all logics) equality does not mean true/false but refers rather to the equality of mapping values. Admittedly, this is a common error for computer scientists to make.
I also am concerned by this notion of "finite definition" which seems to me to be too informal as stated by Juan-Esteban.
Of course, English language is context-dependent. Even formal language is not context-free. For instance, in topology the term "limit" denotes a point which eventually contains a sequence. By contrast, in categorical algebra the term "limit" denotes an object uniquely determined by some kind of commutative diagrams, for instance, products or disjoint unions. Curly brackets {} can be interpreted as a set or class container or the delimiters of Christofell' s symbol. Both delimiters [ ] are used to denote equivalence classes, closed intervals or the numeric function that forgets mantissas.
In logics, a finite word-sequence is a phrase; a phrase having a meaning is a sentence; a sentence which can be true, false or undecidable is a statement: By contrast, in computer science a statement is a command even a declaration; while the true or false nature are only applied to boolean expressions.
An algebra of any domain really should be context free. Differences of definitions and semantics between languages, language properties, must be clearly stated in any formal language grammar before it can be accepted, though it is true that history shows us that any such a language may prove useful long before this point and be maintained, for awhile, on this basis alone. Terminology differences between domains is a different matter, one subject to refinement.
There is a distinction today between languages that deal with structure (mathematics) and languages that deal with relations (logic) - you must either be careful not to confuse the two or careful to state how they are the same. At all costs one must not confuse the two by overloading concepts such as equivalence/equality.
Incidentally, on initial inspection I found your Cantor paper above to be very interesting. I will be pleased to have the opportunity to look at the full length version with proofs.
Dear Steven,
Every mathematical paper contains always some definitions and notations introduced by the corresponding author. In general, these definitions must be only considered in the paper context. Unfortunately, there are a limited symbol set to be used, and this fact oblige us to term different objects by the same symbols. This inconvenient does not matter whenever the author takes care of defining them.
The great french mathematician Henri Poincaré says: "Mathematics is the art of denoting different things by the same name". Of course, he was thinking in equivalence classes and analogies. Analogies are also particular cases of equivalences.
The father of normed spaces, Banach, wrote the following:
A mathematician is a person who can find analogies between theorems;
a better mathematician is one who can see analogies between proofs
and the best mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one who can
see analogies between analogies.
(Stefan Banach 1892 - 1945)
Best regards.
Juan Esteban
P.S. I have sent you my paper about Cantor's theorem via e-mail.
What do you think about this equivalence Stefan = Steven = Esteban?
Thank you for sending me your paper Juan Esteban. Unfortunately, in my experience, many mathematics papers are unreadable because they rely upon transitory conventions, the graffiti and names of dead mathematics, and private language.
Every mathematical statement, be it a definition, theorem or proof. can in principle be written out formally using a finite number of symbols. The set of all such finite statements is countably infinite. However there an uncountably infinite number of real numbers. Therefore there must be some real numbers that cannot be described using mathematical symbols. Of course I cannot describe one of them.
These ideas are used in the Lowenheim-Skolem theorem, that proves for instance there is a countable model for the real numbers.
Note by the way that pi is defined as 6L where L is the sum of the infinite series whose nth term is (-(-1)^n)/n. So when suitably written out formally this would be a description of pi using a finite number of terms.
Due to work of J.P.Jones, D.Sato, H.Wada and D.Wiens the set of primes is identical with the set of positive values of the polynomial:
(k+2){1-[wz + h + j - q]^2-[(gk + 2g + k + 1} (h+j) + h - z]^2
-[2n + p + q + z - e]^2 - [16(k + 1)^3 (k+2)(n+1)^2 + 1 - f^2]^2
-[e^3(e+2)(a+1)^2+1-o^2]^2-[(a^2+1)y^2 + 1 - x^2]^2-[16r^2y^4(a^2-1)+1-u^2]^2
-[((a+u^2(u^2-a))^2-1](n+4dy)^2 + 1 -(x+cu)^2]^2-[n+l+v-y]^2-[(a^2 - 1)l^2 + 1 - m^2]^2
-[ai + k + 1 - l - i]^2 - [p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) -m]^2 - [q + y(a - p - 1)
+s(2ap + 2a - p^2 - 2p - 2) - x]^2 - [z + pl(a - p) + t(2ap - p^2 - 1)-pm]^2}
as the variables a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z range over natural numbers 0,1,2,3,...
Some weeks ago Konrad answered this question: See Chaitin's constant in
http://en.wikipedia.org/wiki/Chaitin's_constant
Dear Hanspeter, "Ich freue mich":
The called Gödel numbering function G sends every finite statement into a positive integer, which is called the Gödel's number of the corresponding statement. If there is a finite statement defining each real number, then one can associated the corresponding Gödel number to each real number. Since every Gödel number is an integer, the consequence is that the real number set is countable; which contradicts Cantor's theorem. Thus, the assumption that every real number can be defined by means of a finite statement is false.
See. http://en.wikipedia.org/wiki/G%C3%B6del_numbering
Of course, the answer is no. I have found this proof while I was following this thread and due to your clever questions.
Thanks for provoking my wit to emerge.
With the help of a German beer, and listening to Bach's music (Die Kunst der Fuge), I closed my eyes and saw that, to find a counterexample , only needed two Pythagorean numbers. Accordingly, if in the fraction (m+1)/m the variable m stands for 16/9, then square root of
(m+1)/m = (16/9+1)/(16/9) = 25/9
is the rational number 5/3.
However, by virtue of Fermats's Last Theorem, higher roots must be irrational.
Another interesting question is the following one.
Given an arbitrary large number m, does there exists another one n such that the interval [n, n+m] contains one and only one prime number?
From my page http://www.researchgate.net/profile/Juan-Esteban_Palomar_Tarancon/ you can download a paper of mine about Cantor's theorem, accepted for publication in IJOPCM, in which you can see that the "polytheism" introduced by Cantor can be refused, although his theorem can be true.
I rather mean "size", because cardinality is defined as the existence of bijections, and a definition is only a convention. Hume's principle, identifies size and cardinality, and it is this principle what it is used to interpret the impossibility of defining a bijection between N and its power set (Cantor's theorem). What I mean is that the impossibility of defining a bijection need not imply size inequality, that is, unlike Hume's principle, the concept of size need not be equivalent to the cardinality notion. In fact, the existence of a bijection implies size equality, but the converse implication need not be true, unless it is proved.
Assuming Hume's principle one can interpret that there are several infinite size or "several divinities" (polytheism), as I have said in a colloquial style.
Finally, I must say, that by no means my claim is a dogma, since to reject Hume's principle, I have said that it must be proved. My only contribution consists of distinguishing between both concepts and providing a criterion to this aim.
I suppose that the existence of real numbers which cannot be defined by finite statements must cause some perplexity. One can reject the existence of invisible men, because he has never seen one. Analogously, one can reject the existence of a number which cannot be defined, because he has never read a definition of it.
definitions as algorithms must terminate whether sets are infinite or finite.
is the question about a finite number of algorithms to define real numbers or not?
I agree with Hanspeter Guettinger that real numbers are dead.
physicists think that what is real isn't mathematical.mathematicians think otherwise.
there is no proof in physics whereas there is no truth in mathematics.
what is reality?
Dear Hanspeter,
I agree.
The symbol dx only can mean a quantity as small as you want, but not a concrete quantity.
I think that quantum mechanics provided us with a gate to go out of this inconvenient. There are indivisible quantities or quanta. An alternative, can be a fuzzy region as Heisenberg principle can imply.
Of course, incommensurable magnitudes exist in our mind, but quantum mechanics induces some doubt..
Dear John Gabriel
When you say that pi exists, are you talking about real world? If it is the case, you are under scope of physics.
Can you accept that you can build a circle formed by points? That is to say, one can split a piece of of physical matter in an endless process? Quantum mechanics negates such a possibility. When reducing the radius of a ball sufficiently, the Heisenberg's indetermination principle says that you get into the kingdom of fuzzy logic in which you not be sure of anything.
Dear Hanspeter,
The links you have cited above deal with classical mechanics. According to Ehrenfest Theorem, classical mechanic laws only deal with expectation values, while actual values run according to quantum mechanics and under the indetermination principle.
Consider that when a particle P is inside a ball of radius R, you know the position of P with an error smaller than R, therefore the error in the measure of the corresponding momentum tends toward infinity as R converges to 0. Thus,
the speed of the particle tends also towards the infinity, because the error in the momentum measure also converges to infinity. In addition, the concept of trajectory cannot be applied to quantum particles.
Likewise if one knows the energy value without any error, it is not possible to determine the time in which such energy occurs, and so on. The path of an electron consists of the average positions of its fuzzy presence, which reach whole space (Schrödinger).
Appendix:
http://en.wikipedia.org/wiki/Ehrenfest_theorem
http://www.amazon.com/Mathematical-Foundations-Quantum-Mechanics-Neumann/dp/0691028931
Of course, but I am talking about real world. By contrast, mathematical constructions belong to mental world.
Mathematical thought can fly freely under those axioms you prefer. However, preferred axioms need not fit into the real world structure. It is experimentation the only authority to accept or reject them.
Dear Hanspeter,
If your contribution must be regarded as brain storming, I must say that you have a good brain. It is an enriching experience dialoguing with you,
"Is there a finite definition for every real number?"
If by "finite definition" you mean algorithm, then Church-Turing Thesis plus Halting Problem shows that the answer is no.
Given the example "pi is defined as the ratio of diameter to circumference" I assume that the question excludes the Church-Turing thesis. The root of 2, for example, can be defined formally as n.n = 2 or "the number that when multiplied by itself is 2." One might prefer n = 2 / n but, actually, this reveals a problem in the use of "finite definition" in that the algebraic definition is not finite while the informal english is.
At least, this is how I interpret the use of "definition" in the question since it does not say anything about the finite nature of the computation.
John,
I can certainly accept this view concerning numbers, and it is well stated.
However, in terms of definitions and the spirit of the question, I think it is fine to leave the mathematical object defined by inference as in the case of n = 2 / n, because it reveals something about the structure of the problem. n in this case is only a number because 2 is, it derives its type from the constant and the operator semantics, otherwise the equation itself is brought into question - and that too is a useful fact. As is the consequent n^3 = 2n.
:-)
Incidentally, it is my view that incommensurable magnitudes are the product only of accepting a universal metric in mathematics.
Well, I think that is time to re-reading work of Julius Wilhelm Richard Dedekind.
Dedekind's cuts are not dead to me and I can work with real numbers, all real numbers, thinking on cuts.
"The idea behind a cut is that an irrational number divides the rational numbers into two classes (sets), with all the members of one class (upper) being strictly greater than all the members of the other (lower) class. For example, the square root of 2 puts all the negative numbers and the numbers whose squares are less than 2 into the lower class, and the positive numbers whose squares are greater than 2 into the upper class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers");[1] in modern terminology, Vollständigkeit, completeness." Wikipedia.
Carlos, ... the conventional approach assumes that it is reasonable to place irrational numbers on the number line in the first place ...
Of course sqrt(2) can be defined as the "length" of the diagonal of a square whose side is 1. An (ideal) square in the plane can be constructed in a finite number of steps. But as John said "one has to be careful to distinguish between a magnitude and a number".
Steven, can you explain please what do you mean by "universal metric"?
By the assumption of a "universal metric" I mean the assumption that relative measures must universally use the same basis. That is, if you take a measuring rod to be the basis of all measure in the system. It's particularly problematic in the cartesian system, it's why you get irrational measures.
The impossibility of stating a finite definition for every real number is equivalent to the impossibility of defining every infinite subset of N by means of a finite statement set; where N stands for the set of positive integers.
To see this fact, consider that you can define a bijection from the power set P(N) of N into the unit interval [0,1] as follows.
Let S be a subset of N and assign the binary number 0.c1c2c3… to it according to the following convention. If 1 belongs to S then c1 = 1, otherwise c1 = 0; if 2 belongs to S the c2=1, otherwise c2 = 0 and so on. Indeed we have stated a one-to-one correspondence between P(N) and [0,1].
Now, define the members of S by means of an infinite set of statements. For instance a member of S is the age of my cousin Mary, another could be the "true" age of my cousin Mary; another the smallest prime greater than the distance from the Earth to the Moon; another the month of the year in which you will accept that not every real number can be defined by means of a finite statement, and so on. Once the infinite set of statements defining the members of S are chosen, you can find the corresponding binary number in [0,1].
Do you think that it is always possible to define any subset S of N by a finite set of statements?
Consider that since you can build an infinite set of predicates P={p1, p2,...} arbitrarily, each of which defining a positive integer, you can only prove that the result must be always equivalent to a finite one provided that you can find a finite predicate set implying all members of P in spite of how the members of P are chosen.
I have twice written the proof based on the Gödel's numbering function. To reject my proof you need to reject Gödel's method together with Cantor's Theorem. If so, I am sure that you deserve a monument.
If we want a serious dialog, at least, you cannot reject my claim without pointing out why, that is, without proving that my proof based on Gödel's numbers is wrong.
This topic deals with real numbers and not with magnitudes. In spite of being incommensurable, the number π can be determined by means of a finite expression. For instance as 3*arcos(0.5). Consider that definition means any expression determining it. Science is not constructed using the ideas of a smart man, but the proofs of any poor man.
If Cantor's Theorem is wrong excellent!!. it is easy to prove. The only you needs is to define a bijection from N into its power set P(N); where N denotes the positive integer set.
I am looking forward to hear from you....
Steven,
I suspect that something much more general is going on here. There is an old paper by N. G. de Bruijn of T. U. Eindhoven titled "On the roles of types in Mathematics" (just google it, it is in pdf form). There the author claims that maths based on ZFC are problematic and difficult to learn because everything is one-typed, namely everything is a set. An integral, a number and a geodesic, all are sets. The author claims that it is more natural to assume that mathematical objects are not of one type, just as in programming languages: there we have the Integer type, the Real type, the File type etc.
So perhaps the circumference and the diameter of a circle are different data types. When you try to compare them you get a straightforward infinity (the infinite number of digits of pi). And infinity is nature's gentle way to tell us that we are talking nonsense.
Dear John Gabriel,
Something puzzles me here, about this claim that there is no number without magnitude.
Consider the following definition of natural numbers that uses only (finite) sets:
0 := {}
1 := {0} = {{}}
2 := {0, 1} = {{}, {{}}}
3 := {0, 1, 2} = {{}, {{}}. {{}, {{}}}}
... etc.
and what puzzles me here is that we haven't considered any magnitudes prior to defining these ?
We don't need the empty set we can simply take any object to start with, let this object be an apple.
We can now define the naturals again as:
0 := apple
1 := {0} = {apple}
2 := {0, 1} = {apple, {apple}}
3 := {0, 1, 2} = {apple, {apple}, {apple, {apple}}}
... etc.
we can do arithmetic with these, just by doing set operations (on finite sets), so I really don't see the need for any measurement to do this ?
If you think that cannot exist numbers without magnitudes it is because there is always an isomorphism between R and any set of magnitudes. In quantum mechanics it is supposed an isomorphism between magnitudes and matrices, even between magnitudes and some kind of operators. However, the number concept was defined centuries before defining the matrix concept.
In abstract algebra, a field being isomorphic to the set of real or complex numbers can be defined without assigning a meaning to each member, even without stating a one-to-one correspondence with the real number set. Isomorphic means with the same structure and the same properties. You can define a three member group G={a,b,c} without assigning a meaning to each of the symbols a, b and c. Of course this group must be isomorphic to Z_3 = {0,1,2}, because there exists only one three-member group. Nevertheless, this isomorphism can be disregarded to define and analyze G. Even ignoring the existence of such an isomorphism, you can find out all structure properties of G.
A child cannot express how old he is without using his fingers, but we are old enough to be able of handling numbers without magnitudes.
Dear Konrad,
of course the von Neumann definition of the naturals does not mention "size" at all. However we do not know what a "number" is. Follow me: The von Neumann definition presupposes formal set theory, ZF. Now, ZF is built on the language which presupposes first order predicate calculus which in turn presupposes an informal set theory (a la Halmos), which in turn presupposes ... fingers for counting!
- Badges
- Science topic
- Similar topics
- Mathematics