Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So infâinf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So infâinf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by â-â), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations infâinf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 â 2 = 1.
In Sewellâs words:
âIn classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of âremovingâ a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real worldâ.
âThere is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about âremovingâ infinite subsets from infinite sets of physical objects. On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to âremoveâ an infinite subset of real objects from an infinite set of them. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as weâve seen, so does the logic of infinite sets in the real worldâ.
âOn the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to âremoveâ an infinite subset of real objects from an infinite set of them.â
This doesnât follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things arenât well-defined unless you specify them in enough detail, but that doesnât make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
Thereâs also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that itâs not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). Itâs of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but thereâs no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesnât follow by necessity from the finite case.
Thereâs also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that itâs not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities).
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/â2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/âmorality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
This doesnât follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = infâinf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 â infâinf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = infâinf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 â infâinf = inf.
So there is a contradiction:
D: C1 ^ C2 â 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R â n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.
Long story short, Sewell:
Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So infâinf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So infâinf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by â-â), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations infâinf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 â 2 = 1.
In Sewellâs words:
âIn classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of âremovingâ a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real worldâ.
âThere is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about âremovingâ infinite subsets from infinite sets of physical objects. On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to âremoveâ an infinite subset of real objects from an infinite set of them. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as weâve seen, so does the logic of infinite sets in the real worldâ.
My reply here has some further context.
Sure, I trust your decisions regarding your time. Thanks for the discussion!
âOn the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to âremoveâ an infinite subset of real objects from an infinite set of them.â
This doesnât follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things arenât well-defined unless you specify them in enough detail, but that doesnât make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
Thereâs also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that itâs not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). Itâs of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but thereâs no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesnât follow by necessity from the finite case.
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/â2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/âmorality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = infâinf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 â infâinf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = infâinf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 â infâinf = inf.
So there is a contradiction:
D: C1 ^ C2 â 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R â n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.