Math's existence

[punkforchrist]

A popular misconception is that there is only one type of mathematics. There are in fact infinitely many varieties. To reiterate what I said earlier, we intentionally chose to use some of the types that are most applicable to nature. Math generally is not applicable to nature, however.

I’ll give an example. The field we are most accustomed to is the field of real numbers that we all learned a little about in high school. But it’s possible to define different and simpler fields. Assume the existence of two numbers, 0 and 1, and two operations, addition and multiplication, defined such that 0+0=1+1=0, 0+1=1+0=1, 01=10=00=0, 11=1. These numbers and operations satisfy the requirements for a field, but good luck finding a non-trivial use for this field in nature.

Is that really a different type of mathematics, or is it just mathematics greatly simplified? When you say that mathematics is not generally applicable to nature, that has me perplexed. The fundamental forces of nature - gravity, electromagnetism, and the strong and weak atomic forces - are indeed expressed by mathematical formulas.

I do agree, though, that there are some different types of mathematics. I’ve already mentioned Euclidean and non-Euclidean geometries as examples. Still, it doesn’t seem necessary that the universe had to have any law-like behavior capable of mathematical expression.

 

[Oreoracle]

Is that really a different type of mathematics, or is it just mathematics greatly simplified?

It certainly is simpler in many senses, but it’s different as well. 1+1=0 is false in the real numbers, but true for this simpler field. There are also fields (namely, those of characteristic 2) in which (x+y)^2=x^2+y^2 while (x+y)^2=x^2+2xy+y^2 in the field of real numbers.

If it still bothers you that every example thus far has been simpler than the reals, then consider the hyperreals. The set of hyperreals contains every real but also every number “infinitely close” to each real number. This gives us unusual properties, the most useful of which allow us to perform calculus without using limits (note that the hyperreals were invented expressly for this purpose).

When you say that mathematics is not generally applicable to nature, that has me perplexed. The fundamental forces of nature - gravity, electromagnetism, and the strong and weak atomic forces - are indeed expressed by mathematical formulas.

As the example of hyperreals illustrates, such phenomenon can be modeled with reals or with hyperreals, using two different versions of calculus. The truth is not fixed in math, it is based entirely on the axioms and definitions you use.

As for not generally being applicable, I mean that we have to choose our axioms very carefully in order for a number system to permit the measurements and calculations science requires.

I do agree, though, that there are some different types of mathematics. I’ve already mentioned Euclidean and non-Euclidean geometries as examples. Still, it doesn’t seem necessary that the universe had to have any law-like behavior capable of mathematical expression.

It seems to me that if the universe behaves in a consistent way then its behavior is necessarily mathematical, since math is just logic equipped with set theory. I suppose we could have a universe that doesn’t operate according to rules, but then it would be fundamentally unpredictable.

 

[punkforchrist]

Oreoracle, you’re the mathematician, so I’ll defer to you on questions of math. What interests me most, though, is when you state:

Oreoracle:

It seems to me that if the universe behaves in a consistent way then its behavior is necessarily mathematical, since math is just logic equipped with set theory. I suppose we could have a universe that doesn’t operate according to rules, but then it would be fundamentally unpredictable.

I agree with this. However, it raises an important question: if the mathematically-expressed forces of nature are not the result of necessity, then are they the result of chance or design? I can’t think of a fourth alternative, and it seems obvious to me that they cannot be the result of chance. If one were to win the lottery once, it would be chalked up to luck/chance. However, if one were to win the lottery a thousand times in a row, then there’s no way chance would be invoked as an explanation.

 

[Oreoracle]

However, it raises an important question: if the mathematically-expressed forces of nature are not the result of necessity, then are they the result of chance or design?

I wasn’t making any claim about the metaphysical necessity of the laws of physics. I was only saying that if there are in fact laws of physics, then they can be expressed mathematically.

I can’t think of a fourth alternative, and it seems obvious to me that they cannot be the result of chance. If one were to win the lottery once, it would be chalked up to luck/chance. However, if one were to win the lottery a thousand times in a row, then there’s no way chance would be invoked as an explanation.

I know you’re going for an analogy here, but I think it’s very misleading to ascribe probabilities to laws of physics. Only events have probabilities. The laws didn’t “happen”, they just are.

Also, could we not easily imagine a universe with different laws of physics? If there is such a large variety of possible sets of physical laws, then this particular set of laws we’re familiar with needn’t be special. It’s true that different laws of physics could preclude life from arising, but then why is life special? Of course we think it is, because we’re lifeforms. But we’re biased.

 

[punkforchrist]

Oreoracle:

I wasn’t making any claim about the metaphysical necessity of the laws of physics. I was only saying that if there are in fact laws of physics, then they can be expressed mathematically.

You had said this in your previous post:

I suppose we could have a universe that doesn’t operate according to rules, but then it would be fundamentally unpredictable.

I interpreted that as saying that the forces of nature are not metaphysically necessary. Did I misunderstand you?

I know you’re going for an analogy here, but I think it’s very misleading to ascribe probabilities to laws of physics. Only events have probabilities. The laws didn’t “happen”, they just are.

So the forces of nature are without any explanation whatsoever? The forces of nature are instantiated in events, so I think it’s perfectly reasonable to talk about the probability that they will be instantiated again in the future. Keep in mind the problem of induction.

Also, could we not easily imagine a universe with different laws of physics? If there is such a large variety of possible sets of physical laws, then this particular set of laws we’re familiar with needn’t be special. It’s true that different laws of physics could preclude life from arising, but then why is life special? Of course we think it is, because we’re lifeforms. But we’re biased.

I’m not focusing on life or the fine-tuning of the universe’s initial conditions for the possibility of life. Rather, I’m suggesting that the fact that there are any forces of nature at all is indicative of cosmic design. Sure, the universe could have been imbued with different forces of nature, but it also could have been imbued with none whatsoever.

 

[Jim_Baur]

I am the OP.

I found an answer.

en.wikipedia.org/wiki/Stephen_Hawking

Hawking has stated that he is “not religious in the normal sense” and he believes that “the universe is governed by the laws of science. The laws may have been decreed by God, but God does not intervene to break the laws.”[285] In an interview published in The Guardian, Hawking regarded the concept of Heaven as a myth, believing that there is “no heaven or afterlife” and that such a notion was a “fairy story for people afraid of the dark.”[146]

 

[mikekle]

Why not? There is no obvious reason why this should be the only universe. There may be others which are utterly different.

This is true. I often find it comical when I hear people talking about mathematics being universal, no matter where you go in the entire universe, these people think something man created could be universal LOL

I tend to think there is mathematics and physics on other worlds, but more than likely, MUCH different than our understanding of them, for all we know, on their planet, maybe they can show how 2+2=5. Anything is possible, and I would find it quite illogical to think something man created would be used all thru out the universe.

 

[Oreoracle]

I interpreted that as saying that the forces of nature are not metaphysically necessary. Did I misunderstand you?

Fair point, I guess I just don’t like introducing metaphysical terminology when logic is sufficient. My position is that there is nothing logically contradictory about the notion of a universe free of physical laws (other than a self-referential law like “this universe has no other laws”).

So the forces of nature are without any explanation whatsoever? The forces of nature are instantiated in events, so I think it’s perfectly reasonable to talk about the probability that they will be instantiated again in the future. Keep in mind the problem of induction.

I thought you were only talking about the probability that the laws are what they are, not the probability that they will stay what they are once we’ve got them. As I mentioned earlier, it would be interesting if the laws of physics changed over time, because if they change over time in a predictable manner, then the real laws of physics are those that describe how the former laws change over time. This would resolve the problem of induction if true.

For example, we could say it’s a law that the moon will take a certain amount of time to orbit the Earth. If we took this to be a law, then the law would change over time, since the moon is receding from us a few centimeters a year. The way this “law” would change can be predicted, so it really doesn’t pose a problem for us because we’ll replace it with the “static” law that accounts for the change.

But if such “static” laws do not exist, it seems to me that the universe would be beyond our comprehension unless the appearance of consistency has arisen due to chance.

Sure, the universe could have been imbued with different forces of nature, but it also could have been imbued with none whatsoever.

And this is what I meant when I said that I don’t think the laws “happened”. There needn’t have been any imbuing whatsoever. The universe just was a certain way, just as Christians believe God simply is a certain way.

 

[Oreoracle]

I tend to think there is mathematics and physics on other worlds, but more than likely, MUCH different than our understanding of them, for all we know, on their planet, maybe they can show how 2+2=5.

It is possible that our laws of physics break down somewhere in the universe, but mathematical theorems will always hold. This is because they are based on logic and not experimentation.

Anything is possible, and I would find it quite illogical to think something man created would be used all thru out the universe.

Naturally, aliens would use very different notation and symbols than we do, but the concepts would be the same. Aliens might also develop different subjects of math faster based on their needs.

 

[punkforchrist]

Thanks for the thoughtful reply, Oreoracle.

I take it that, then, that you don’t believe the forces of nature have any explanation whatsoever. This isn’t the same as what theists say about God. We believe God’s existence is explained by a necessity of his own nature.

 

[Oreoracle]

I take it that, then, that you don’t believe the forces of nature have any explanation whatsoever. This isn’t the same as what theists say about God. We believe God’s existence is explained by a necessity of his own nature.

I’ve never seen an argument that establishes God’s necessity without using circular reasoning. The ontological argument is a great example of the circularity I’m talking about: If something is perfect, it must exist. I define God to be perfect, therefore he exists. It’s cheating–you could prove anything like that.

 

[punkforchrist]

I’ve never seen an argument that establishes God’s necessity without using circular reasoning. The ontological argument is a great example of the circularity I’m talking about: If something is perfect, it must exist. I define God to be perfect, therefore he exists. It’s cheating–you could prove anything like that.

I didn’t have the ontological argument in mind. I was thinking about the argument from contingency, the third way, and the modal third way. None of these arguments are circular.

 

[Oreoracle]

I didn’t have the ontological argument in mind. I was thinking about the argument from contingency, the third way, and the modal third way. None of these arguments are circular.

Correct me if I’m wrong, but the argument from contingency seeks to establish the existence of a necessary being. It does not specify what the necessary being is. For example, I could use the argument to “prove” that the universe is necessary, just by substituting “universe” in place of God in the argument’s conclusion.

In order to have the God of Christianity, you need either additional arguments or additional assumptions. Otherwise you have only the most vague conclusion of all time: something exists, and it has to exist, but we don’t know what it is.

 

[punkforchrist]

Correct me if I’m wrong, but the argument from contingency seeks to establish the existence of a necessary being. It does not specify what the necessary being is. For example, I could use the argument to “prove” that the universe is necessary, just by substituting “universe” in place of God in the argument’s conclusion.

In order to have the God of Christianity, you need either additional arguments or additional assumptions. Otherwise you have only the most vague conclusion of all time: something exists, and it has to exist, but we don’t know what it is.

Cosmological arguments have two parts. The first is to establish that a first cause or necessary being exists, and the second is to demonstrate that this necessary being possesses the divine attributes. Assuming the universe exists contingently, then it would have an external cause, and this cause could only be a timeless, changeless, immaterial, very powerful, and arguably personal agent. That’s how William Lane Craig defends the argument, for example.

 

[weller2]

And then we look at which explanation is the most parsimoniuous. And the winner is…?

Currently, it’s a tie between God and multiverse.

This is because the laws as we know them are too complicated. Philosophically speaking, the lowest level of reality should be as simple and possible. The laws we know now are anything but simple, which makes it highly likely that they are not fundamental – i.e. there are not something that “just exists”. They appear to be either designed (hence God) or randomly selected out of much larger space of possible laws (hence multiverse).

Curiously enough, the God hypothesis seems to currently have more support than the multiverse hypothesis. This is because there is no evidence for multiverse and at the same time there are some results (i.e. the holographic principle, black hole information problem, quantum eraser experiments) which indicate that a necessary component of universe is information. So if the lowest level of reality indeed involves information, then universe is an information processing machine, which would mean that either the universe itself is sentient, or the level “below” the universe is sentient. At this moment you have either pantheism or panentheism, but either way, you arrive at the God concept.

 

[Oreoracle]

The laws we know now are anything but simple, which makes it highly likely that they are not fundamental.

I’ll grant that the laws are hard to understand, but they are simple in the sense that they make few assumptions. For example, in quantum mechanics, particles are allowed to occupy multiple places at once. In fact, “quantum particles” can travel arbitrarily large distances in arbitrarily small amounts of time. That’s very difficult for the average person to grasp, so it is counterintuitive. But it is simple, because it’s basically the smallest assumption one can make about how these particles move. We aren’t assuming that they follow a particular path, or obey a certain speed limit, etc.

Anyway, I think a lot of people confuse “simple” with “intuitive”. But most physicists I’ve read about believe that the ultimate laws of physics, if we ever discover them, will be few in number and will tie everything together nicely. In other words, they will offer the simplest possible theory. If our current theories are even close to the true laws, then I would agree with the physicists. For example, while it’s harder to understand, the theory of relativity makes fewer philosophical assumptions than its predecessor, Newtonian physics.

 

[Ion]

I think God could have created the universe with either a Euclidean or non-Euclidean geometry, for example. I don’t think he could have made 2+2=5, since that would be a logical contradiction.

2+2=5 is a contradiction considering that 2+2=4 and 2+3=5

Do not put God and “can’t” in the same sentece. God is almighty. It is not good to say for example that “God can’t do otherwise than He already said is going to do” because this is in fact our limitation in finity, we lack the comprehension of the infinity of his power.
And the best example is our salvation from death through Jesus.

 

[Bob_Crowley]

I know this is a large, large question.

Where do people like Hawking and Dawkins think the math that we see in nature originated?

THANKS!

Related to this question is that of physical constants ie. those quantities or factors which appear to be built in or immutable as far as this universe is concerned.

Where did they come from?

I copied these from “Fundamental Laws of Physics” by F.W. Constant (funny you should say that).

Gravitational Constant
Coulomb’s law constant
Ampere’s law constant
Speed of light in vacuo.
Planck’s constant
Boltzmann’s constant
Avogradro’s number
Gas Constant
Electric Charge unit.
Faraday’s constant.
Rest mass of electron.
Ratio of mass of proton to mass of electron.

 

[weller2]

I’ll grant that the laws are hard to understand, but they are simple in the sense that they make few assumptions. For example, in quantum mechanics, particles are allowed to occupy multiple places at once.

QM is actually simple if counterintuitive.

A good exaple of simple law is Newton’s laws – you have several simple equations and one magic number.

But take the Standard Model – it is basically one equation and 19 magic numbers. One equation is the good part, 19 magic number is the bad part, because nobody can explain why they have values they do. String theorists attempt to explain them by curvature of space, but the problem is, in this theory there is 10^500 possible ways to bend space, so 19 magic numbers have been traded for a magic space curvature (i.e. a manifold). And still nobody can explain why this manifold here and not that one over there. Multiverse proponents handwave that by saying that all these remaining (10^500)-1 universes actually exist, but are inobservable. Duh.

Now compare that to contruction of basic arithmetics. You start with a set of simple axioms: A proof system for arithmetic and with these you can build any proof, i.e. that 1+1=2: 1+1=2 without invoking any magic values.

 

[Oreoracle]

Now compare that to contruction of basic arithmetics. You start with a set of simple axioms: A proof system for arithmetic and with these you can build any proof, i.e. that 1+1=2: 1+1=2 without invoking any magic values.

But you’re comparing apples and oranges. Math isn’t attempting to “explain” anything, nor is it attempting to predict phenomena. The fact that we use math for those purposes is immaterial, since we could also develop impractical mathematics. Math is a formal science and physics is a natural science–we shouldn’t expect them to be equal in this regard.