Is Mathematics an ideology?

[EmiEmi]

Here is a quote from a person from another forum (he's not replying to me. I am just a lurker) But since I don't know much about mathematics. I wonder if he's right. Is mathematics an ideology just like Democracy, Communism, etc...?

Absolutely false charge. I categorically said that the right to vote is an extension of the principles of basic human rights. That does not indicate that right to vote is the highest of all rights.
Again, false. ideology is any idea that cannot be verified with actual empirical proof. There is no empirical proof that '1' exists, that is what we simply use as an arbitrary point in our reference. If i completely got rid of '1' and devised a fractal, it would make the math complex, but would be no different than changing an ideology.

Everyone agrees that 1+1 =2. That is because everyone assumes and agrees that two whole numbers should be evenly spaced out. There is no empirical basis to that.

Math is NOT a science. Science is defined as the organized knowledge in the form of testable predictions and explanations about the universe. Science is pertaining to any empirically observable fact. Math is not an empirically observable fact, the numbers 1 and 2 don't just hang around in space. They are artificial constructs WE do.

Yet they rely on the ASSUMPTION that the number system is evenly spaced out. There is no empiric reason for having a linear number system, a non-linear number system is simply harder to compute, not wrong. This makes it a belief system, not an empiric fact.

No. math makes a lot more assumptions. You simply are not educated enough in math to see it. To you, math is only what a few numbers and formulas do in real world applications. You have no idea what is imaginary math, math that deals with nothing but math itself, it has ZERO correlation to the quantifiable universe and when you are competent to do math of that level, you may understand the principles outlining math.

You are admitting that logic and ideology (which you have arbitrarily separated on your whim) have a nebulous demarcation.
The very fact that i can define 1 and 2 differently than rest of the world, in their entire concept, and still make all the calculations work,makes 1 and 2 a belief system. They are not empiric, because one cannot define an empiric fact differently. You may use different words, but you will describe the interaction exactly the same for a phenomena. Math is not a phenomena, its strictly an idea. As with any idea, it can be changed from the norm to alternate forms and still applied to the universe.

You didn't prove me wrong. I said that math is a belief system. I can use far greater examples but it will fly over your head, so i am using the most basic one:

We as humans simply, on a leap of faith, have assumed that our number system is such that 1,2,3,4,5,etc. are all evenly spaced out from each other. That is called a linear number system. They are all linearly separated from each other.
It doesn't have to be so. It is obviously, very convenient to have a linear number system. If we had a non-linear number system, where instead of 1,2,3,4,5,etc. we have 1,23,33,49,73 as the whole numbers, it would create non-linear calculations, thus requiring a much bigger formula and referencing to for every calculation. (because now, 23-1 is not equal to 33-23. The decimals separating 1 and 23 are different than decimal spacing of 23 and 33).

You saying that 'everyone agrees 1+1=2' only re-inforces the point that math is the most successful ideology of all-time. Everyone agrees, because it is the logical method that leads to least amount of complexity. It doesn't make it empiric, as the ONLY thing that is correct and its a system that rests on assumption of convenience, not empiric fact.
Again, you are lost, confused and don't know what you are talking about. F=ma is not math, its a physics phenomena. It has empiric evidence. It cannot be changed.
2-1=3-2=4-3 is an assumption. Its not a physics or a material phenomena.
Its an ASSUMPTION.
The very fact that i can do that, means it is an ideology, not a fact. F=ma is fact. Nomatter what words, symbols or equations or base of math i use, F=ma will always hold true in empiric verification.

There is nothing stopping me, except convenience from using a numeral system that is non-linear and it would be the same thing as heresy: one guy with an alternate view that still works from the majority.
Says the guy who knows that the linear number system is an arbitrary assumption, something that you clearly do not have the mathematical knowledge to even grasp.
No, re-defining is not re-naming. Your analogy is false, because you are talking about re-naming the planet. I am not calling 1 and 2 something else. I am discarding the basis of their existence by simply saying that a number system does not have to be linear. if it doesn't have to be linear, then the relation between 1 and 2 does not have to be preserved.
Mass and length are empirical facts. 1 and 2 are our constructs, not empirical facts. Do you even understand the difference between empirical facts and logical constructs ?

An assumption can be valid and still be an assumption. I can assume a number system that goes 1,22,39,45,88 etc. and it would also be just as valid as 1,2,3,4,5 etc. Just more cumbersome.

What you can't get right, is that 1+1=2 relies on the mathematical assumption of a linear number system. A number system is linear out of mathematical convenience, not out of empirical fact. A non-linear number system is exponentially more cumbersome, but also correct.Because if i gave you such an analogy, you would not understand it. But there are parameters that can make an integral unworkable too. Stick to the basics, if you are having such a difficult time understanding that 1+1=2 is an assumption because we assume a linear number system, i don't expect you to understand how parameters to an integral can render it unsolvable or solvable based on the parameters itself.
Because you are defending the legitimacy of a government that exists solely on the basis that it has conquered China (the CPC). There is no basis to their power in any other means, since people's satisfaction levels is unrepresented.In overwhelming majority of the cases, you cannot because you do not have the data pertaining to their choice, nor the data pertaining to the results. In physiology or physical interactions, some things are easy to predict, which your example is.
I never said they are the same in function, but ALL democracies derive their power by mandate of the people. That is where they are all alike and that is the point of their legitimacy, which makes all non democratic governments, illegitimate.

Incorrect. You yet to justify why the comparing aspects are applicable in the first place. Economic performance is not benchmark for legitimacy, otherwise it would be legitimate for me to take over your house, provided i treated it better than you did.The assumption is that the number system is linear. That assumption leads to the definition. If that assumption is removed, the definition, by default, changes.
1+1=2 is not a basic fact, its a logical construct from the assumption that number systems are linear.
Not at all. Your comment pre-supposes a linear hierarchy of rights. Is the right to a fair trial a greater or lesser right than right to work ? Is the right to not be raped greater or lesser than right to not be tortured ? There is no highest of all rights, there are a set of fundamental, inviolable rights, one of them being right to choose one's government.There is no contradiction.
Says the guy who does not even realize that the number system pre-supposes a linear number system.
The numbers do not need to be whole numbers. I am not re-defining the definition, simply pointing out that the whole concept of what numbers are, is an assumption, something that actually is investigated in advanced number theories.
Again, false deduction. Science uses logic. Logic is not science. Go read your definitions again, science is the explanation of something that can be verified as a
phenomena. Math is not a phenomena, it is a concept.
Science uses english too. English is not a science. So science is not science if it uses english, eh ?
True, but the proof of math is in logic of the system. That does not mean that the system doesn't use certain assumptions to base its framework on.
And that definition is a definition of convenience, not empiric. The universe did not give us empiric evidence on why we should use whole numbers, we simply use whole numbers because its convenient.
Not immoral tactics, you are simply refusing to see the point that if i discard whole number system, its no different than discarding a belief system. Nothing changes empirically, we simply re-work our equations. It is not the same as saying F=ma doesn't exist, because that would be discarding a phenomena that the universe provides us empirically.
I will explore the subject further once you wrap your head around the simplest of the assumption math makes, namely the whole number system. its unwise to jump too far ahead. There are many books on number theory that deals with the inherent assumptions made in our numeral system, most of the examples are far more complicated than the most basic one every mathematician knowns, namely, the assumption that is the whole number system,
Explore it yourself. I happen to be a source on the topic. If that doesn't suffice, i instruct you to begin by reading Georg Cantor and his thesis on transfinite numbers.
Incorrect. Logic is any statement that does not violate the rationality of the pre-defined system. Ideology can be logical or illogical, everything can be logical and illogical. I can have an ideology that states that God is infinite, which can be deduced to God also has infinite power. That is a logical statement within the predefined system (God exists). If i say 'God is infinite but i can do better math than God' then that statement would be illogical.
As you can see, logic has nothing to do with the set of assumptions about the system.
You still do not understand.It simply shows that what you and the rest of the world does, is based on an assumption of linear numbers.
What you are doing is changing the definition of the word 'democracy'. I am not changing the definition of '1' (by calling it 5 for e.g.), i am changing the assumption that leads to 1 : that numbers must be counted in whole number system.
That is a fundamental assumption you simply fail to see.Only if we pre-suppose using a linear, whole number system. If i use non linear number system, 3-2 and 2-1 would have different values.
So far all you've shown is personal opinion after personal opinion. I've gone through way worse formulas than simply parroting of opinions.You are parroting the opinion and a flawed one when you call math empiric, a science and the number system to be absolute. None of them are true.
If i assume a non-linear system of numbers, instead of a linear system, then my numbers are not evenly spaced out, thus 3-2 is not equal to 2-1 in that system.
Its a simple concept. Yes, that would be my definition, but we are dealing with an assumption here and if i can make a system work with my own set of assumptions, it proves that what you were using in the prior case is also an assumption.No, i am not. You are confusing between phenomena and ideology. F=ma is a phenomena represented in that form. 1,2,3 are ideologies.
Big difference but i don't expect you to understand, since you have shown no inclination to learn, only parrot your deficiencies in math as some misguided idea of absolute knowledge.The values represented by 1 and 2 are assumed on the basis of a linear whole number system. We don't have to use the whole number system, we simply do it because its convenient. If i don't use the whole number system, then the values assigned to 1 and 2 will change non-linearly, thus making 3-2 and 4-3 two different results.
Again, you confuse between a phenomena (color of the apple) to an abstract ideology (mathematical numbers).
Sure, you can change the definitions of force and acceleration and mass. But the phenomena will not change, meaning the relationship will still be preserved no matter what you come up with and will have to call it something else. That is just calling something a different name.
If i assume that my number system is not based on whole numbers but based on...say transfinite numbers only, then i am not just changing the nomenclature of 1, i am fundamentally changing its value.
And this proves, once and for all, that it is a dogma, a belief, something we just assume- that our numbers have to be whole numbers.
False. We ASSUME that our set of numbers are whole numbers. They don;t have to be.Again, you fail to understand. I am not calling 1 a 13, i am assigning the value of 13 to 1, the value of 33 to 2, as would be the case in a non linear number system.

You still can't grasp that basic concept.
I am definitely not wrong when it comes to this math bit, that much, any mathematician worth his/her salt will tell you in a heartbeat.
It matters greatly whether it is linear or non linear, because in the linear system, 1+1=2. In the non-linear system, there is no necessary logic that the increases in the numbers are consistently incremental, so there is no reason for 1+1 to be 2.

I know what you mean, its just that you don't have a clue to what i mean! The whole world agrees that 1+1=2. That doesn't change the fact that it rests on an ideology, aka the whole number system. We simply dreamt it up due to convinience. There is no universal law dictating that numbers must be counted in equal increments, its just very convenient to do so.

Doesn't change the fact that 1 and 2 are still extrapolations of an assumed system (whole numbers).
*sigh*
listen carefully. i will say this only one more time before i give up to your asinine proclamations on topics you are ill-qualified to talk on.

This is our current number theory:
its called the whole number theory, where every whole number is defined as incrementally greater than its predecessor is to the predecessor's predecessor.
Ie, if we have 3 whole numbers a,b,c who are one after the other, then c-b = b-a.
This is because the whole number uses linear spacing.
What we call these numbers (one, uno, ek,un, etc) is irrelevant.

Now, there is no empiric reason from the universe to have a whole number theory.
I can have a number system, where a,b,c are not equally spaced out from each other. To simplify this (since you are having such a tough time comprehending this example), in the non-linear system, say i assign a,b,c values of 1,4 and 8.
That is how we count, we go 1,4,8,33, etc (this is NOT the same as calling 2 a 4, 3 an 8, 4 a 33. It is assigning the VALUES of 4 to a 2, 8 to a 3, 33 to a 4, etc).

In such a non-linear number system, then c-b is NOT equal to b-a.

False.
Try not to repeat yourself, it doesn't make you any more right.
False. You do not comprehend, that's all.
More repetetive nonsense.Those numbers don't mean much, as those are given by people who have no free will.
yes, it does. The right or wrong of the statement is dependent on how often you can be right vs you can be wrong. In this case, since you do not know the people at hand, most of their choices pertaining to life are not privy to your judgement.You are correct. i have no explanation that would make sense to you, considering that you are failing to grasp the concept of one of the most basic assumptions in mathematics.
Its a similar scenario on how i have no explanation for fluid dynamics to a 5 year old.

Here is a quote from a person from another forum (he's not replying to me. I am just a lurker) But since I don't know much about mathematics. I wonder if he's right. Is mathematics an ideology just like [edited] political ideologies.
All of this argument was started by the concept of a better ideology. If Mathematics has assumptions and be correct - then so can other ideologies". [

Another point was since a lot of higher mathematics proofs and equation have no physical or empirical evidence, then that makes mathematics ideology.

"Not at all, since these rights are used as defining parameters to democracy. They are no different than defining parameters in various other systems that i have given examples of."
"If other systems can have defining parameters, so can a political system."

 

[Drakkith]

A word of warning to anyone posting, please limit your discussion to the mathematical part of this and refrain from going off on a tangent about any political/philosophical aspects which may/may not be correct or that are opinion based.

The original question is: "Is mathematics an ideology just like Democracy, Communism, etc...?"

 

[EmiEmi]

I don't usually bother with posts like these because they're all nonsense to me, but one part did catch my eye

Everyone agrees that 1+1 =2. That is because everyone assumes and agrees that two whole numbers should be evenly spaced out. There is no empirical basis to that.

This is wrong, and I won't care to explain why.

To the person who posted this, please do explain why. I know it sounds stupid, but please. Thank you.

 

[FactChecker]

There are very fundamental characteristics of mathematics that distinguish it from an ideology:

1) The basic assumptions of mathematics were slowly developed from the need to do basic counting and calculations. A study of abstract algebra or plane geometry show that the assumptions are very basic and natural. There is nothing mythical about them and, in fact, it would be difficult to imagine alternatives. (Many alternatives mentioned in the OP for a different number system would not really do that. They would only change the names or symbols used in mathematics.)

2) Mathematics is used all the time to make predictions that are verified to be true. Every time a calculation of the area of a square is compared with the measured result, the calculation turns out to be true. So math has been used to predict results trillions of times without a counterexample.

3) Mathematics reacts to ANY counterexample that is discovered. And the reaction can take the form of a change to its very foundational assumptions. In pure math, the assumptions are not contradicted by any known examples. Even the most basic assumptions are open to question and revision if a counterexample is found.

These are not characteristics of an ideology.

 

[FactChecker]

To the person who posted this, please do explain why. I know it sounds stupid, but please. Thank you.

That depends on your definition of addition. You can define addition so that 1+1=2 when looked at in one setting, but not in other settings. Suppose addition is defined as adding measurements around the Equator, and 1 = the distance half way around the Earth. Then 1 + 1 gets you all the way around the Earth's Equator and back to where you started. So 1 + 1 = 0. In that abstract algebra, there is no number 2.

 

[EmiEmi]

That depends on your definition of addition. You can define addition so that 1+1=2 when looked at in one setting, but not in other settings. Suppose addition is defined as adding measurements around the Equator, and 1 = the distance half way around the Earth. Then 1 + 1 gets you all the way around the Earth's Equator and back to where you started. So 1 + 1 = 0. In that abstract algebra, there is no number 2.

That is because everyone assumes and agrees that two whole numbers should be evenly spaced out. There is no empirical basis to that.

What about this part?

 

[FactChecker]

If you are interested in fundamental questions like this, you might be interested in abstract algebra. Suppose you define '1' as something and '+' is defined. Then '1+1=2' says that '2' is the result of '+'ing '1' with itself. Addition might represent adding distances end to end in a straight line on a plane. But it might represent something else entirely (rotations of an object, chemical reactions, etc.). But in all cases, the symbol '1' represents something and '1+1' means that you are adding two of the same thing. If the numbers represent distances, that translates to adding two of the same distance together. The result is denoted by the symbol '2'.

 

[pbuk]

I haven't read much of this, I didn't get beyond "That is because everyone assumes and agrees that two whole numbers should be evenly spaced out. There is no empirical basis to that" which is the exact opposite of the truth.

"1" is not an invented concept, it is something that we observe. When I look at you, I can see 1 person. If I have a collection of similar objects in front of me I can divide them into groups. If I keep dividing the groups until I can't divide them any more, each group will contain exactly 1 object.

If I pick up 1 of the objects, I have 1 object in my hand. If I pick up 1 more object I have 1 more than 1 object in my hand, and we assign the label "2" to the number of objects in my hand. If I pick up 1 more object then I have 1 more than 2 objects in my hand and we assign the label "3" to the number of objects, and so on. In this way we can observe that whole numbers are evenly spaced out, it is not an assumption.

Edit: I don't know about the "other forum" but I'm pretty sure that reposting wholesale from PhysicsForums without permission is a breach of copyright, so if you were thinking of doing that, don't.

 

[FactChecker]

I think that the short answer is that the equal spacing is true because of the definitions.
Suppose '1' represents a distance, and '1' + '1' represents the sum of two equal distances '1' placed end-to-end in a straight line on a plane. That distance is represented by the symbol '2'. By those definitions, '2' is always twice the distance of '1'.

There can be other definitions in other geometries where distance is not additive. You can 'add' distances around a circle and end up with a zero total distance.

 

[EmiEmi]

What about this part?

Edit: I don't know about the "other forum" but I'm pretty sure that reposting wholesale from PhysicsForums without permission is a breach of copyright, so if you were thinking of doing that, don't.

I lurk and I don't post there. So no worries about copying your posts. I just wanted to understand what they mean and I am not sure what they mean and what better than you guys, I thought.

 

[Nidum]

Is it not true that whilst mathematics can be based on any arbitrary number system and methology whatever is actually being described, quantified or analysed is actually always the same thing deep down ? Therefore use whatever number system and methodology you like but any description , quantification or analysis can always be mapped back into the conventional simple system ?

Most mathematics is axiom based . Mathematics based on axioms chosen is seen to work . Axioms are shown to be valid .

Man has always sought to understand and quantify his environment and has always striven to understand and quantify in the simplest way .

 

[pbuk]

Is it not true that whilst mathematics can be based on any arbitrary number system

No. The mathematics of whole numbers is based on (an abstraction of) the observation I described, which is clearly invariant and not arbitrary.

Most mathematics is axiom based . Mathematics based on axioms chosen is seen to work . Axioms are shown to be valid .

Axioms are not "shown to be valid": they are the things which we ASSUME to be valid in order to show that other statements are valid.

 

[HallsofIvy]

I lurk and I don't post there. So no worries about copying your posts. I just wanted to understand what they mean and I am not sure what they mean and what better than you guys, I thought.

That's not the point. You did copy a great deal fro the other forum and post it here! Did you have permission to do that?

Another point was since a lot of higher mathematics proofs and equation have no physical or empirical evidence, then that makes mathematics ideology.

As far as "ideology" is concerned, mathematics is not an "ideology" because no mathematician is saying "this is true" but rather "if this is true then ...". All mathematics statements, even if they are not written in that form.

 

[Ssnow]

No, I think it is not an ideology. In front of an idea you can believe it or not and you should rely only on feelings. In front of a theorem you can know whether it is true or false.

 

[EmiEmi]

That's not the point. You did copy a great deal fro the other forum and post it here! Did you have permission to do that?

I just read the rules since I lurk. Yes, it is fine. I don't see anything saying no as long as you don't use it as your own work.

It ain't no capital crime, but you should avoid it. There are lawyers everywhere. And it is bad manners in any case.

Yea, I lurk so I don't really pay attention to any of it.

I'll post the original link of the history forum, but I avoid it in case it provokes a thread close-down since it involves politics as a mod here warned me. No politics on math forums. Don't read it so this thread doesn't get entangled with politics. I'm just curious about the explanation of the math.
http://historum.com/asian-history/92795-who-would-you-have-supported-1962-sino-indian-war.html
I was also embarassed at my leap from a non-science history forum to a hard math one. It would probably make posters here have a good guffaw and stop posting serious responses, so please ignore it.

 

[FactChecker]

A lot of what is in the original post that you copied is mistaken. 2 - 1 = 3 - 2 is true because of how the symbols of 1, 2, 3 and + are defined. They are not assumptions.

 

[FactChecker]

Yes. It is fine.

It ain't no capital crime, but you should avoid it. There are lawyers everywhere. And it is bad manners in any case.

 

[EmiEmi]

A lot of what is in the original post that you copied is mistaken. 2 - 1 = 3 - 2 is true because of how the symbols of 1, 2, 3 and + are defined. They are not assumptions.

No, re-defining is not re-naming. Your analogy is false, because you are talking about re-naming the planet. I am not calling 1 and 2 something else. I am discarding the basis of their existence by simply saying that a number system does not have to be linear. if it doesn't have to be linear, then the relation between 1 and 2 does not have to be preserved.
Mass and length are empirical facts. 1 and 2 are our constructs, not empirical facts. Do you even understand the difference between empirical facts and logical constructs ?
incorrect. As simple as that. If S is a set of non-linear numbers, then x' is not equal to y'.
End of story.
1+1=2 is an assumption based on a linear number system. No amount of psuedo-science from you will change that.

Part of the OP's quote
Isn't the linear system of numbers an assumption and that's what he's attacking here?

 

[FactChecker]

Part of the OP's quote
Isn't the linear system of numbers an assumption and that's what he's attacking here?

The number system is an algebra with an operation '+'. It can be used in many ways. Some applications have a 'linear' meaning on a straight line. Others do not. Here is (roughly from vague memory) a way to start building up a number system and it's properties that is not based on ideology:

Suppose we have a set of elements and an operation '+':between any two elements
1) If there is one element which leaves all elements unchanged when it is added, we will denote it by '0'. So e+0 = e for all elements e
2) If each element, e, has another element that, when added, gives 0, we will denote that by -e. So e+(-e) = 0 for all elements e.
3) If there is one element which can generate all other elements by repeatedly adding or subtracting it, we will denote it by '1'. So 1, -1, 1+1, -1-1. 1+1+1, .-1-1-1,... are all the elements.
4) If two things are equal, and the same operation is performed on both, then the results are equal.

With those conditions satisfied, let's define the symbols '2'=1+1; '3'=1+1+1; ...
Then a lot of what your quote questions is forced to be true. It is not 'ideology'.

A lot of what your quote says is assumed as ideology is really not assumed. They are postulated. There is a difference. When one studies Euclidean geometry, some things are postulated and certain results are obtained. But those things are not postulated in non-Euclidean geometry, and the results are different.

This is the field of abstract algebra and it is very well developed and formalized. I am afraid that I am not doing it justice, working from memory.

 

[EmiEmi]

The number system is an algebra with an operation '+'. It can be used in many ways. Some applications have a 'linear' meaning on a straight line. Others do not. Here is (roughly from vague memory) a way to start building up a number system and it's properties that is not based on ideology:

Suppose we have a set of elements and an operation '+':between any two elements
1) If there is one element which leaves all elements unchanged when it is added, we will denote it by '0'. So e+0 = e for all elements e
2) If each element, e, has another element that, when added, gives 0, we will denote that by -e. So e+(-e) = 0 for all elements e.
3) If there is one element which can generate all other elements by repeatedly adding or subtracting it, we will denote it by '1'. So 1, -1, 1+1, -1-1. 1+1+1, .-1-1-1,... are all the elements.
4) If two things are equal, and the same operation is performed on both, then the results are equal.

With those conditions satisfied, let's define the symbols '2'=1+1; '3'=1+1+1; ...
Then a lot of what your quote questions is forced to be true. It is not 'ideology'.

A lot of what your quote says is assumed as ideology is really not assumed. They are postulated. There is a difference. When one studies Euclidean geometry, some things are postulated and certain results are obtained. But those things are not postulated in non-Euclidean geometry, and the results are different.

This is the field of abstract algebra and it is very well developed and formalized. I am afraid that I am not doing it justice, working from memory.

The definition of postulate (from Google) is "suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief."

It sounds like an assumption, if not what is the difference in mathematics?

But if you postulate A, and get Aa result and you postulate B and get Bb result - what exactly does this mean? Doesn't this mean the mathematics are not true and universally applied?

 

[FactChecker]

With a postulate, you admit that it is something you can not prove now, but you are seeing what you can conclude from it. There are examples of mathematical (and physical) theories where contradictory postulates are adopted one at a time to see what can be concluded from each. Because both cases are studied, neither is assumed to be universally true. For instance, the Euclidean and non-Eucludean Geometries of two types lead to 3 different theories. Euclidean geometry has Euclids parallel postulate and the others do not. The physics associated with flat space versus curved space results from the different geometries.

 

[homeomorphic]

It seems like the guy might be implying that math is an ideology in the sense that you can't trust conclusions that use math. But you can trust them, provided that the math is actually a good model for what you are trying to model with it, and that can be established empirically.

But you don't always have to formally test your results to be confident of them in every case. For example, if there you know 5 women are coming to your party and 8 men, then you can predict that you will need 8+5=13 seats in order to accommodate them. In this example, you already have enough life experience to have tested the model informally to know that it works.

Also, math doesn't claim that anything is true directly. It says that under such and such assumptions, then so and so follows from that. It's a way of getting conclusions from starting assumptions, not getting starting assumptions from nowhere. As such, there's no ideology.

However, we can never really be sure that the theorems of math are really true, due to Godel's incompleteness theorem, which states that if you have enough power in your axioms to handle arithmetic, there is no way to prove the the consistency of the axioms, unless you assume more unproven axioms.

An example of some inconsistent axioms would be

1) The moon is blue.
2) The moon is not blue.

Axiom 2 contradicts axioms 1 very obviously in this case. Once you prove a statement to be both true and false, the whole thing falls apart because then you can prove that anything is both true and flase. You could also have a situation where instead of a direct contradiction, you can do some deductions and then arrive at a contradiction. What Godel's theorem is telling us is that we can't be absolutely sure that we haven't done that with our present foundations of math, although the contradiction would have to be very non-obvious, since we haven't found any. So, really, it's only "empirically" true that math is consistent. The conclusion is not that math is an ideology, but that it's on the same footing as other sciences, in the sense that absolute proof is not actually possible, contrary to what some people might have thought about it.

 

[Dale]

@EmiEmi

If you would like to continue this discussion, please do so in the context of your own questions and reasoning, not the assertion of a non participating third party. It makes discussion problematic if we don't know if we are responding to a question you have or just some irrelevant text. Please see the rules regarding third party conversations.

This thread is closed.