# On the philosophy of analysis

• Atreya

#### Atreya

Hello people!

My name is Atreya and I'm 18 years old. I'm currently studying for my A levels. I used to love physics and mathematics a lot, and I still do... sort of. A couple of months back, I came across a big philosophical problem. I'd like to name this problem analytical nihilism. This problem destroyed my love for physics and mathematics and actually for all subjects involving analysis. It has devastated me in a way and I'm writing here to seek your help in solving this problem.

I’ll explain the dilemma through an example first and then I’ll give you the general picture. Back in the days, Isaac Newton discovered his Law of Gravitation by performing experiments on certain masses which were at certain distances apart. But the fact is that he only performed the experiments on certain and a finite number of masses and only at certain and a finite number of distances. How could he be so sure that his law will hold for all masses at all distances?

One may say he could perform more experiments to extend his proof, but it is the nature of real numbers and real variables to be infinite, so how can he go through all possible variables in the finite amount of time that he has on Earth?

This idea extends to all things in science. How can we be sure of any form of analysis we do when analysis is infinite but we are finite? How can we reconcile this meaninglessness?

At the back of my head, I know intuitively that the universe is beautiful and that this ugly picture must be wrong and that there must be some order to the universe. But I just can't seem to find a logical, non-intuitive answer. It's killing me.

I hope someone will be able to give me a solid explanation to reconcile this problem and to help me see the well-ordered beauty of the universe.

Thank you.

You seem to have difficulty distinguishing between "physics" and "mathematics". Yes, Newton based his theory of gravity on a finite number of examples- and, in fact, it was later shown that he was not completely correct. General relativity explained observations that Newton's theory could not. And there is no reason to think that General relativity is a "perfect" law of gravity- no physical theory is perfect. Hopefully, we are getting closer and closer to "the truth" although we know we will never get to "perfect truth". Why do you have such a problem with that?

Mathematical theories, on the other hand, are defined by us. They have the properties we put in them. Although we may discover surprising results we did not know were in the theory ("emergent properties") we still know that they are there because of our definition. While it is certainly possible for a given proof to be wrong, once a correct proof is given, the result cannot be later falsified.

Science doesn't deal in truth -- it deals mainly in (empirical) evidence. Nor does science prove theories -- it decides between alternatives.

Newton gathered enough evidence to infer his theory of gravitation was much, much better than any of the alternatives.

P.S. IIRC Newton's evidence involved celestial bodies, not weights in a laboratory. (although I imagine some data was gathered by the attraction of masses to the Earth) I don't believe they didn't have tools sensitive enough back then to measure the gravitational attraction between laboratory masses.

I hope someone will be able to give me a solid explanation to reconcile this problem
Are you sure anything really need reconciling? What is it that you enjoy about math and science and stuff? Are you sure the stuff you enjoy has anything to do with this philosophical notion you're worried about?

My problem with the "physics" aspect is that if we can never attain a perfect theory, what's the point of even studying physics? You say we get closer and closer to the "truth", but in reality we don't because the truth is inherent in the infinite and we can never comprehend or so called "reach" the infinite, no matter how hard we try.

And sorry for my mistake there. I completely agree with you that pure mathematics is defined by us. However, the applications are not. My problem is with the applications of mathematics to the real world. It's the same idea. We can justify the applicability of Mathematics to the physical world for a certain experiment at a certain point in time and in a certain region in space. But how is it that we so confidently justify our physical theories for all space and time?

To Hurkyl,

I beg to differ on that issue. Science is a human need. It is driven by human curiosity to understand our world, and to understand what the world is made up of and how it works. That is in some sense, the "truth". Experiments are means to get closer to that truth.

I enjoy studying mathematics and physics because I feel that they help me understand my world better. The problem that I defined earlier, which I believe is philosophical, is impeding my enjoyment and enlightenment because it makes me feel that everything I knew or understood before is false. That's why I need someone to help me reconcile this, to show me that our understanding of the world is not false.

My problem with the "physics" aspect is that if we can never attain a perfect theory ...
I enjoy studying mathematics and physics because I feel that they help me understand my world better.
Do you see the mismatch between the two? You say the problem is that you can't have perfect knowledge. But when you said what you enjoy, it wasn't "I enjoy attaining perfect knowledge", but that you enjoy bettering your knowledge...

To Hurkyl:

Thank you for pointing that out. But I think you are missing the issue at hand. What I wrote about enjoyment is about me and my feelings towards these subjects.

Yes, I enjoy building up knowledge, as I'm sure you do too. And here I am, building up knowledge on what a perfect theory is and how our present day theories relate to the perfect theory. Hopefully you'll be able to offer something on that.

Or perhaps you meant something which I didn't catch?

To Hurkyl,

I beg to differ on that issue. Science is a human need. It is driven by human curiosity to understand our world, and to understand what the world is made up of and how it works. That is in some sense, the "truth". Experiments are means to get closer to that truth.

I enjoy studying mathematics and physics because I feel that they help me understand my world better. The problem that I defined earlier, which I believe is philosophical, is impeding my enjoyment and enlightenment because it makes me feel that everything I knew or understood before is false. That's why I need someone to help me reconcile this, to show me that our understanding of the world is not false.

Physics doesn't deal with the 'truth' in absolute (human) terms. It deals with defining theories and tools which empirically approximate the physical universe which may be an enlightning, enjoyable, exercise with pragmatic consequences. (While at the same time, sometimes debunking older believes.)

I think if you want to understand your feeling, or the world, you are better of with psychology and philosophy [, maybe theology]. (If that is what really drives you.)

Dear Atreya,
Mathematics lies in the pure realm you are thinking of. Godel proved conclusively that it is impossible to ever 'know' mathematics completely. It will always be incomplete.
To one degree or another all scientific models are derived from underlying mathematical principles. The measurement problem you bring up, in relation to Quantum Mechanics is a brick wall you can only see through one 'bit' at a time with all other 'bits' of information of a particular moment being necessarily fuzzy.
We don't justify physical theories, we use them until a better theory is arrived at. The thing about all subsequent theories is that their models are necessarily consistent with the previous model's results but to some degree more refined. When, over a span of time, all measurements are in agreement with the model and none are consistently in disagreement with the model, the likelyhood that the model works in out sector of the universe is good. It would be inaccurate to say that any model is true or false. It is just a model, to the degree that it works it is useful. For instance many calculations scientists use apply Newton's formula and not GR because the difference, in some instances, is significantly less than the errors arising from accuracy of measurements they are being applied to.
Our understanding of the universe is incomplete. There is still lots more to discover.
I find beauty, even in our incomplete models.
mathal

That's why I need someone to help me reconcile this, to show me that our understanding of the world is not false.

Sadly, in the above sentence, there are 2 things that are not well understood - "someone" and "world". Oh, the term "understand" is also poorly understood.

You probably know this well, but i am going to say it anyway because you just've registered here. Instead of seeking fundamental knowledge, science uses a more pragmatical approach. It builds a framework of common-sense assumptions and axioms(self-evident truths, don't laugh, they are supposed to exist) and lays a structure that aims to model the world.

Philosophy has been known for nurturing a plethora of great thinkers through the centuries - you don't have to stick to science if you feel it's not getting you closer to truths(if you think they exist, though philosophers tend to have much fewer answers than scientific models are able to propose). Religion would be another option, as long as you don't discuss it here.

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Thanks so much for your replies. I'm going through the material and will try to fathom whatever I can.

I still feel this question is scientific in nature because it strikes the core assumptions and axioms of science. But I also realize that science itself cannot provide an answer to this question because this question is beyond empirical logic. It is something more intuitive in nature.

To Maui: I've also posted this question on a philosophy forum and I received very eye-opening replies. Just for the sake of it, I might post them here. But anyways, I realize that religion is also important in answering this question. But of course, I won't bring that discussion here.

I still feel this question is scientific in nature because it strikes the core assumptions and axioms of science.
I am not aware of scientific axioms. Could you tell me what they are? Or you may be talking about the scientific method? You will find axioms in mathematics.

Have you taken a look also at the http://en.wikipedia.org/wiki/Problem_of_induction" [Broken] regarding science, not mathematics?

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I am not aware of scientific axioms. Could you tell me what they are? Or you may be talking about the scientific method? You will find axioms in mathematics.

Here is the commonly accepted definition of axiom:

"In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered to be either self-evident, or subject to necessary decision. In other words, an axiom is a logical statement that is assumed to be true. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived"

http://en.wikipedia.org/wiki/Axiom

And here is a statement that is considered to be a self-evident axiomatic truth in science(without it science would turn into a chronology):

People and scientists act out of their free will, and not of predetermined events in the past

Here is another one:

"Ours is not a simulated universe"

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Thanks so much for your replies. I'm going through the material and will try to fathom whatever I can.

I still feel this question is scientific in nature because it strikes the core assumptions and axioms of science. But I also realize that science itself cannot provide an answer to this question because this question is beyond empirical logic. It is something more intuitive in nature.

To Maui: I've also posted this question on a philosophy forum and I received very eye-opening replies. Just for the sake of it, I might post them here. But anyways, I realize that religion is also important in answering this question. But of course, I won't bring that discussion here.

With question I suppose you meant: How can we be sure of any form of analysis we do when analysis is infinite but we are finite? How can we reconcile this meaninglessness?

Darling, philosophy will tell you that there are no certainties, except for that you can only accept you exist, and probably other people exist too. What you do with that is up to you.

Physics will tell you a lot about the nature of things, and is beautiful. If you like that, go study it, but don't expect a lot of philosophical answers.

Psychology will tell you a lot about how people work as 'apes,' but doesn't provide any absolute truths in the philosophical sense.

Philosophy will tell you a lot about how to ask existential questions, and will give you lots of (historical) meaningful theories. But don't expect a lot of answers which make sense to you as a human.

Best thing for you is to find out what you're interested and comfortable with. Best wishes on that.

And here is a statement that is considered to be a self-evident axiomatic truth in science(without it science would turn into a chronology):

People and scientists act out of their free will, and not of predetermined events in the past

Here is another one:

"Ours is not a simulated universe"
Thank you for posting that here.

I would not consider, "People and scientists act out of their free will, and not of predetermined events in the past" a self evident truth. There are a lot of questionable words used in there. Such as, "free will" and "predestined".

Or the second one. Universal claims like, X is something that is not part of the universe" are hard to verify or make sense of.

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Thank you for posting that here.

I would not consider, "People and scientists act out of their free will, and not of predetermined events in the past" a self evident truth. There are a lot of questionable words used in there. Such as, "free will" and "predestined".

Or the second one. Universal claims like, X is something that is not part of the universe" are hard to verify or make sense of.

One has to start from somewhere and being born naked and clueless in an african savanna certainly hasn't helped help the situation with establishing the basics.
Of course, you are free to doubt anything. Or rather you were not free but forced to doubt everything by events in the past over which you had no control. Or whatever.
Surely there is at least one self-evident axiomatic truth in life - I exist(there is a personal experience)

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My problem with the "physics" aspect is that if we can never attain a perfect theory, what's the point of even studying physics? You say we get closer and closer to the "truth", but in reality we don't because the truth is inherent in the infinite and we can never comprehend or so called "reach" the infinite, no matter how hard we try.

Although to some it might seem like clutching at straws, I always found the following thought particulary pleasing, so I wanted to share it with you in the light of the quoted passage:

although it is indeed impossible to ever be sure your theory is correct, it is not impossible to ever attain it :)

Yeah Mr. Vodka, that is an interesting thought! We will never know when exactly our theories are perfect, so for all we know they could be perfect now?

And to Willowz, axioms are everywhere is science! Axioms are in fact all of our laws. There is no way to prove a law for all space and time. So we accept the laws as universal, or as self-evident truths; that if a few experiments show a certain trend under certain conditions, then all experiments should do the same. So Newton's law of gravitation is an axiom. So is the conservation of charge and the conservation of relativistic energy. All the laws basically.

Another fundamental axiom in science as well as in every single thing we do as human beings, intelligent human beings that is, is that nature is uniform (uniformity of nature in technical terms). It means that every experiment or observation represents an accurate sample of the universe as a whole and regularities in the experiment will show up as regularities in the universe. This assumption allows us to make the quantum leap from experiment to universal theory. It is such an intuitive assumption that we simply cannot live without it. Even as I explain this axiom to you, I am implicitly assuming the uniformity of nature; that since this explanation that I am partly imitating from an article I read was able to convince me, it should be able to convince you since nature (the logic you and I possess) is uniform.

When you start to see the assumptions we take for granted in just living our lives, you start to see that both human life and science are implicitly a matter of faith! But we don't realize this because the faith we have in the uniformity of nature (UN) is so darn strong, it's almost ingrained inside us! Check out this article for a better development of these ideas: http://www.princeton.edu/~grosen/puc/phi203/induction.html

I think it would extremely hard to draw out axioms in science. Or I am misinformed.

I mean you can call something an axiom in a given field. But, then you would have to outline where this axioms starts failing. Or else you would have to give all encompassing axioms. Sorts of theories of everything. Axioms are self evident by definition. But, scientific theories aren't apparent by definition.

Here are scientific postulates that I have so far found.

Sixteen Postulates of Science

1. Science is logical (utilizing the appropriate form of logic), reasonable, and rational.

2. Science makes well-defined claims based upon the best available evidence.

3. Scientific hypothesis must be falsifable.

4. Scientific experiments should be repeatable under similar circumstances.

5. Science requires that claims be examined by qualified peers.

6. Science views unexplained gaps in theories or evidence with suspicion.

7. Science requires caution both in performing experiments, and in examining and evaluating evidence.

8. Science requires efforts at objectivity, both in control of variables and of biases.

9. Science does not accept coincidence or unlinked or unproven correlations as proofs.

10. Science does not accept undocumented anecdotal evidence as good proof by itself.

11. Science demands extraordinarily good evidence for extraordinary unconventional claims.

12. Scientific favors parsimony: that the simplest adequate explanation is preferred.

13. Science assumes that the Laws of Nature are universal, if relative.

14. Science demands the honest use of the scientific method and truthful reports.

15. Science demands every effort be made to control or assess all variables.

16. Science needs the uninhibited exchange of ideas and greatest possible discourse of the material.
From http://www.unl.edu/rhames/courses/current/sciencegoals.htm" [Broken].

Note 13.

Axioms have an inherent sense of absolute'ism. Science or scientific theories do not. Deriving axioms from theories seems pointless.

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Hi Atreya

You might find David Deutsch's latest book very helpful, The Beginning of Infinity. In it he further develops his concept of explanatory principles. I think this might be what you are in search of! It is knowledge we are in search of in science, and this knowledge applies more or less universally, depending on the subject. As he demonstrates, the reach of some kinds of explanations, such as those in physics, is infinite.

Best wishes.

Your post reminds me of Freud in his early papers describing neurotic patients with his Theory of Sexuality. He basically said "These are my findings on the matters, and I worked with 18 patients. Perhaps if someone worked on a 19th, they might find that I'm wrong," and "but then again, **** it, I'm betting my *** on this" he did not say, but totally could.