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# Hypothesis vs. theory, First person experience vs. objective science

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So, let me ask the following question.

If someone (who did not have access to the rest of science) claimed that 2+2=4 would this be a hypothesis or theory?

Deductive reasoning would suggest that an infinite number of experiments would result in reinforcement of this belief. Do we need to test that 2+2=4 in order to label this as something stronger than a hypothesis?

If so, what do we do when dealing with math that cannot be tested? IE how to treat the infinity, or even what happens when we deal with really large numbers?

Obviously deductive reasoning is often used in place of experimental data, in every discipline of science.

So where EXACTLY is the limit on how much deductive reasoning we can use?

If I am a master of deductive reasoning I might be able to be 100% sure of something that would cause problems for others. Am I then able to label more of my conclusions as theories rather than hypotheses without testing them?

If a person can have an understanding of a subject that they are as certain of (which of course only they can know) as a scientist is of the math he does not experimentally test and yet uses to draw conclusions from his research, then are they entitled to call their conclusions well tested theories?

There seems to be a fundamental contradiction here in the concept of the scientific method.

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According to Kline in his "Loss of Certainty", the logical validity of the statement two times two equals four (somewhat different from two plus two but. . . . ), was only proven in the late nineteenth century, in fact just before the major defactioning of maths occassioned by the consequences of Cantor's ideas. The results of Godel, Skolem and Fitch, in the twentieth century followed this, illustrating that human thought is not up to the full task. We, as humans, have our situation to deal with, the rest is pretentious, and while fun, is primarilly, head wanking.

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Though I do not doubt someone might have said that, that does not make it so. The only thing that was lacking was an understanding of how we know that 2 x 2 = 4 expressed in written form. From what you say it is apparently still lacking in written literature.

It is as follows - The concept of multiplication is actually more natural and an immediate consequence of numbers than addition.

If refer to the number one, I must realize that one is an attribute of something else, like an adjetive. It has no meaning on its own. You have one apple, one orange, one basketball. You cannot have a one.

You can however have a one, of a one, of a basketball. You can have two one basket balls. You can have two two basketballs, which of course gives you four basketballs.

This is pure deductive reasoning. All you must do is take 2 images of one basket ball, then put them together to get what you then label 2 basketballs. Then you simply take the image of two basketballs and make two of it the same way you doubled all contents of the previous image. This new image would contain 4 one basket balls within it.

Other mathematical truths can be deduced in a similar manner, and people do so on a regular basis. It's just hard to translate this reasoning into words, since it deals mostly with manipulating images.

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One times one equals one, but the result is a square whereas the components were lines. One times one times one is a cube, one times one times one times one, you tell me.

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Those are unecessary spatial metaphors though... visualizing one times one as a square is not necessary to show that having one one basketball results in having one basketball.

You can multiply one times one times one times one all day long without needing to come up with some spatial metaphor to attribute to it... There are only 3 dimensions in real life space, so therefore the spatial metaphor must end there.

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Not at all. There is no point multiplying one by one if the result is imperceptible. The resulting uncertainty about the condition of our initial one, is the point.

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If someone (who did not have access to the rest of science) claimed that 2+2=4 would this be a hypothesis or theory?

...

There seems to be a fundamental contradiction here in the concept of the scientific method.

You point at something which I find very interesting, and which took me a while to realize. I have read numerous popularized accounts on mathematics (like Paul Hoffman's biography on Paul Erdöz, The man who loved only numbers, or Simon Singh's The Code Book).

2+2=4 is a construct and assumes that you are using a numeric base in which the construct constitutes a true statement.

In base 2, there is no such thing as 2+2 (it would read 10+10=100)

In base 3, 2+2 = 11

In base 4, 2+2 = 10

For bases 5 and above, 2+2=4 will always be true.

In other words, the linear number system (0, 1, 2, 3, 4 etc) are constructed based on our counting system and we prefer a base 10 system.

This means that seen from a *semantic* point of view, 2+2=4 is limited to be expressed only in certain situations.

If we accept that the statement represents a true statement (which it does as long as we agree on the values of 2 and 4), then it applies to everything in the universe. However, while we can predict that it holds true in all cases, we cannot guarantee it because we cannot know. We could disagree on the values - or the values could be uncertain, as in quantum physics - and therefore the statement 2+2=4 could be an approximation.

Counting is therefore just a practical and very basic way to use mathematics to explain something. It is, however, not in contradiction with the scientific method because the scientific method requires that we agree on the value of all variables.

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All numbers are approximations, this is demonstrated by, inter alia, incommensurability and the proofs that 0.99 recurring equals one.

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Not at all. There is no point multiplying one by one if the result is imperceptible. The resulting uncertainty about the condition of our initial one, is the point.

I don't understand... If you say you have one group of one basketballs how is the result impreceptible without using a square metaphor? I am not labeling the basketball a metaphor because it is what you actually have one of.

You point at something which I find very interesting, and which took me a while to realize. I have read numerous popularized accounts on mathematics (like Paul Hoffman's biography on Paul Erdöz, The man who loved only numbers, or Simon Singh's The Code Book).

2+2=4 is a construct and assumes that you are using a numeric base in which the construct constitutes a true statement.

In base 2, there is no such thing as 2+2 (it would read 10+10=100)

In base 3, 2+2 = 11

In base 4, 2+2 = 10

For bases 5 and above, 2+2=4 will always be true.

In other words, the linear number system (0, 1, 2, 3, 4 etc) are constructed based on our counting system and we prefer a base 10 system.

This means that seen from a *semantic* point of view, 2+2=4 is limited to be expressed only in certain situations.

If we accept that the statement represents a true statement (which it does as long as we agree on the values of 2 and 4), then it applies to everything in the universe. However, while we can predict that it holds true in all cases, we cannot guarantee it because we cannot know. We could disagree on the values - or the values could be uncertain, as in quantum physics - and therefore the statement 2+2=4 could be an approximation.

Counting is therefore just a practical and very basic way to use mathematics to explain something. It is, however, not in contradiction with the scientific method because the scientific method requires that we agree on the value of all variables.

Well the question about science I was talking about was that if you agree that you can use mathematics to prove something, and mathematics can be shown to simply be deductive reasoning, and you can also show that deductive reasoning is used in many other ways during the course of science (to bridge the gap between experimental data and conclusion) then isn't it true that the same hypothesis can become a theory using different balances of testing and deductive reasoning/mathematics?

And if so, where is the line drawn after which the balance is too dependent on reasoning to be considered scientific? How many parts of the claim (that can supposedly be supported using deductive reasoning or math) have to be tested before a hypothesis becomes a theory? And/or where is the line drawn after which the claim/theory is considered questionable simply because of too much reasoning?

I think the answer is that there is no such line. I think that there is always enough reasoning being used in making a conclusion that a person who is not careful or not experienced in dealing with reason itself can make a mistake. And I think that someone really proficient at deductive reasoning can make a theory based on deductive reasoning and common experience that is better tested than any other theory.

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...2+2=4 is a construct and assumes that you are using a numeric base in which the construct constitutes a true statement.

In base 2, there is no such thing as 2+2 (it would read 10+10=100)

In base 3, 2+2 = 11

In base 4, 2+2 = 10

For bases 5 and above, 2+2=4 will always be true.

In other words, the linear number system (0, 1, 2, 3, 4 etc) are constructed based on our counting system and we prefer a base 10 system.

This means that seen from a *semantic* point of view, 2+2=4 is limited to be expressed only in certain situations.

I have a few questions about your comments. How would predictions of quantum mechanics change if all equations were in base 2, or base 3, or base 4 ? Does anyone use base 2, base 3, or base 4 mathematics to search for mathematics of quantum gravity ?
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Well the question about science I was talking about was that if you agree that you can use mathematics to prove something, and mathematics can be shown to simply be deductive reasoning, and you can also show that deductive reasoning is used in many other ways during the course of science (to bridge the gap between experimental data and conclusion) then isn't it true that the same hypothesis can become a theory using different balances of testing and deductive reasoning/mathematics?

The distinction between "hypothesis" and "theory" seems an artificial one. If you mean "can we use math to test a theory," then I believe the answer is no. Math guarantees that certain conclusions follow deductively from a given idea. In all cases, experiment makes the final decision.

-Will

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If someone (who did not have access to the rest of science) claimed that 2+2=4 would this be a hypothesis or theory?
Neither. “2+2=4” is a mathematical proposition, not an assertion about objectively reality. It can’t be experimentally reproduced.

The modern view of Math is that it mostly concerns formalism. Though useful to the point of being indispensable to Science, it isn’t itself a science, a subtle but important distinction (to which I’ve not done justice in this brief post, but, for brevity’s sake, don’t intend to).

If so, what do we do when dealing with math that cannot be tested? IE how to treat the infinity, or even what happens when we deal with really large numbers?
We rely on formalism – that is, follow the specific rules of the specific mathematical formalism being used – and don’t worry about that, as Korzybski famously said, the map (Math) is not the territory (objective reality).
If a person can have an understanding of a subject that they are as certain of (which of course only they can know) as a scientist is of the math he does not experimentally test and yet uses to draw conclusions from his research, then are they entitled to call their conclusions well tested theories?
No. Hypotheses obtained from mathematical formalism may be called promising, and/or that highest mathematical superlative, elegant, but until they can be tested by some sort of measurement of objective reality, they can’t be called well tested theories. If no experimental test is practically possible, or worse, conceivable, it’s arguable that it should even be called a scientific hypothesis. Examples of well-known scientific ideas in this category are the many worlds interpretation and string theory.
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How would predictions of quantum mechanics change if all equations were in base 2, or base 3, or base 4?
They do not change at all, regardless of the numeral system used. Our preference for base 10 is almost certainly a relic of the anatomical fact that we have 10 digits on our hands, so first learned to count to 10.
Does anyone use base 2, base 3, or base 4 mathematics to search for mathematics of quantum gravity?
Since the introduction of modern electronic computers and calculators, practically everyone uses base 2, though we’re usually aware of it only when some artifact, such as a rounding error – reminds us of it.

It’s important to understand the distinction between a number, and the numeral used to represent it. Numbers, even ones that have no exact finite common numeral representation, are exact, and independent of numeration systems. Thus, seeming contradictions such as ughaibu notes

All numbers are approximations, this is demonstrated by, inter alia, incommensurability and the proofs that 0.99 recurring equals one.
demonstrate not that the number “one” is approximate, but that it can be represented by many numerals, including $1$, $0.99 \ \mathrm{recurring}$, $0.\overline9$, $0.999 . . .$, and $0.\overline1_2$.
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I have a few questions about your comments. How would predictions of quantum mechanics change if all equations were in base 2, or base 3, or base 4 ? Does anyone use base 2, base 3, or base 4 mathematics to search for mathematics of quantum gravity ?

My point was merely that the tautology 2+2=4 is accepted as true because it is true within the realm of mathematics. It is however not a *true sentence* unless the values of 2 and 4 can be accepted (as Craig points out, it is a matter of agreement).

Exactly what you study has no bearing on the mathematics, it's rather the other way round: How you choose to express your numbers has a bearing on the results. Thus 2+2=10 is confusing unless you know we're in base 4.

The physical reality of 2+2=4 is something else, entirely. That 2+2=4 is true is because all integers (expect) can be expressed as the sum of two smaller integers. Mathematical purists will say this reflects a "truth" in the universe, whereas I am of the opinion that it is merely an interpretation of what we observe.

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The distinction between "hypothesis" and "theory" seems an artificial one. If you mean "can we use math to test a theory," then I believe the answer is no. Math guarantees that certain conclusions follow deductively from a given idea. In all cases, experiment makes the final decision.

-Will

So then, to use another thread topic for an example, bell's inequality must be verified by experiment? This makes no sense, since if bell's inequality must be verified by experiment, it can not be stated that it must be satisfied in order for a local hidden variable theory to be true in the realm of quantum mechanics. Instead it could simply be the case that the inequality wouldn't hold anyways in a local hidden variable model, since we did not experimentally verify that it did. (And supposedly never could, since you claim there is no such thing)

You can not say that it was verified outside the realm of quantum mechanics and then try to apply it to QM, because that is not randomly sampling from the population you wish to make a prediction for.

Rather you are relying on deductive reasoning to show that the inequality should hold if there was a local hidden variable theory.

And additionally you are always relying on reasoning to explain that an infinite number of assumptions in any experiment are trivial in nature. (Like for example that in a kinematics experiment, an object doesn't change mass during the course of the experiment)

Finally, it seems to me that experimental results are never capable of producing conclusions by themselves. Rather someone draws the conclusions (using reason) based on the results. You might try to say that all their reasoning has all been verified experimentally (even saying that everyone everywhere has always seen that A=A 100% of the time) but then anyone can do the same.

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So then, to use another thread topic for an example, bell's inequality must be verified by experiment?

Indeed, the experiments to show that reality violates Bell's inequality are generally callled Aspect experiments after the first person to perform such an entanglement experiment. We have had long discussions on Bell in the past.

Unless of course you are asking (and it seems that you are, though I didn't originally understand) if you need to experimentally test deductive logic. I would think that a philosopher would be aware that if deductive logic breaks down, we have no tools left to work with. The fundamental assumption of both philosophy and physics is the validity of deductive logic.

However, deductive logic always follows from an initial idea. Hence, the deductive logic ties the idea to the conclusions we can draw from that idea.

-Will

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They do not change at all, regardless of the numeral system used. Our preference for base 10 is almost certainly a relic of the anatomical fact that we have 10 digits on our hands, so first learned to count to 10.

I have to question this in a very serious way. I would ask what happens if we use a non-standard numeration system, like Unary?

Also what happens if things like 0, $\emptyset$, and $\infty$ are rejected as NaN?

Does not that change the outcome? I would expect that if the rules of computation were changed then the predictions would likewise change, even accross base ten to base ten systems. You change the rules, you change the outcome, correct? Or am I missing something here?

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