Questioning the supremacy of mathematical formalism in understanding the universe

[ittiandro]

I have become increasingly interested in physics in recent years, more specifically in the Special and General Relativity.

Unfortunately, my mathematical background, limited to college algebra, does not allow me to fully understand some fundamental concepts, especially those of curved space and gravity, at the core of the General Relativity.

While admitting that my mathematical skills and perhaps my intelligence are a part of the problem, certain questions have arisen in my mind regarding the role of mathematics as the main and sole paradigm of understanding nature, when applied to advanced physics and cosmology. What is the nature of the understanding that physicists and cosmologists claim ( or strive ) to attain through their mathematical constructs, impervious, as they are, to conceptual representation?

This questioning arises from my belief that mathematics, when applied to explain reality, i.e. outside the sphere of pure mathematics, are a language and, as such, they must have an object, a term of reference, distinct from (and not coterminous with) the mathematical entities or formal logical connections through which the language articulates itself.

It is kind of odd that the mathematical formulations all start, as they did with Einstein, from a questioning at a conceptual level ( indeed scientists wouldn’t embark on the mathematical path if they didn’t have a question to answer) but then the initial conceptual reference is lost sight of under the increasingly abstract maze of symbolic mathematical constructs, which become themselves the object of (and the answer to) our query.

By losing sight of the conceptual formulation of the problem underlying our quest, we perhaps unwittingly reduce its object to the outer mathematical shell and in so doing we risk reducing reality to a creation of our mind, the subjective expression of its inner workings.
Emmanuel Kant built a bridge between idealism and realism by holding that, if all our knowledge BEGINS with experience, ultimately our understanding of the world is inescapably shaped by the A PRIORI structures of our mind, the categories. To the extent that the formal logical connections of the mathematical constructs are, I believe, akin to these A PRIORI categories of our mind,( a mere reflection, as it were, of its inner structure) the supremacy of mathematical formalism in advanced physics and cosmology, would signify the lapse into idealism, which would run counter the very essence of science and its quest for objectivity and certitude.

Mathematics are essential in laying out complex problems and providing solutions with a maximum of precision, in a way that our common language is not able to, but shouldn't we able to bring to light at least the essential conceptual thread underlying the mathematical sequences and their conclusions? When a physicist, at the end of an extenuating mathematical voyage through G.R. finally " understands" concepts such as curved space, gravity and the like, I’d indeed love to enter his or her mind to see what he or she understands, beyond the usual (and indeed beautiful) logical tightness of the mathematical equations.

In Special Relativity, in spite of the counter-intuitiveness of the implications of Space and Time , our mind is not beyond some degree of conceptual understanding, because traveling at the speed of light, although not within our experience,is still within our conceptual grasp, in a way that curved space is NOT: not only is CURVED SPACE not given in “ sensible intuition” , as Kant would say, but we don’t even know, intuitively and conceptually, what it means for space to be curved. However logically coherent the mathematical explanations may be, are they sufficient to establish the existence of such a mathematically defined space,ontologically? Sillogisms can be logically true, too, but their conclusions wrong if the premisses are false !

I have a feeling of " deja vu" on this, when I remember Kant speaking of the Antitheses and Paralogisms arising when Reason makes judgments " beyond the manifold of sensible intuition".

The British philosopher A.J.Ayer, in the wake of logical positivism and linguistic analysis once gave the " coup de grace" to metaphysics, when he unabashedly declared meaningless( sic !) any statement about objects whose existence is not empirically verifiable. (I don't know what he would think of'' any assertion about curved space...).

Sometimes I think that our employment ( or perhaps over-employment?) of mathematics and our acceptance of strictly mathematically defined entities such as CURVED SPACE and the like, do bring metaphysics back into science , which may run counter the very essence of science.



Perhaps the only way out of my quandary would be to be assured that notions such as curved space and the related notion of gravity, however conceptually elusive, have been shown to have predictive validity.



May be somebody can comment on this.

Ittiandro

[Philosophy Explorer]

Just to say, it would help if you broke up your post into paragraphs.

Welcome aboard.

PhilX

[wtf]

Perhaps the only way out of my quandary would be to be assured that notions such as curved space and the related notion of gravity, however conceptually elusive, have been shown to have predictive validity.

The confirmation of Einstein's theory of gravity by the observations taken during the solar eclipse of 1919 is one of the most famous science experiments in history.

The findings made Einstein a celebrity overnight, and precipitated the eventual triumph of general relativity over classical Newtonian physics.

http://www.wired.com/2009/05/dayintech_0529/

There have been many experimental verifications of modern physics. It's not just an abstract mathematical theory. Reality works the way the equations say it does; at least to the limits of our ability to measure, and until the next scientific revolution.

[ittiandro]

Perhaps the only way out of my quandary would be to be assured that notions such as curved space and the related notion of gravity, however conceptually elusive, have been shown to have predictive validity.

The confirmation of Einstein's theory of gravity by the observations taken during the solar eclipse of 1919 is one of the most famous science experiments in history.

The findings made Einstein a celebrity overnight, and precipitated the eventual triumph of general relativity over classical Newtonian physics.

http://www.wired.com/2009/05/dayintech_0529/

There have been many experimental verifications of modern physics. It's not just an abstract mathematical theory. Reality works the way the equations say it does; at least to the limits of our ability to measure, and until the next scientific revolution.

I do not question that Einstein’s theories have been EMPIRICALLY VALIDATED . Indeed, from a practical,scientific standpoint, mine is perhaps a moot question. I nevertheless think that, if there is, as I believe, a difference between SCIENCE and PHILOSOPHY of science, my question is philosophically relevant.

That a scientific hypothesis is empirically validated , still makes me want to know where and how, from inside THE STRING of mathematical equations we can infer, CONCEPTUALLY, the vindication that nature behaves according to the formal order of the mathematical statements, unless we deem that this answer is coterminous with their internal logic .
It is a bit like an encrypted statement where the original letters have been replaced by other letters, obeying a code or a key.

Here, too, even before the key is discovered, there is an internal order ( supplied by the key) which the letters follow as they combine in words , but the meaning of the encrypted statement is neither the key itself nor the logical order that the encrypted letters follow in combining according to the key. This meaning is what emerges and is CONCEPTUALLY translatable and understandable in our everyday language.

This goes back to my initial premiss, which, of course, I am quite open to see challenged that
1) mathematics are a language and that
2) the object of this language is “ meta-mathematical”. i.e. it is not the language itself and its symbols . Only by challenging these presuppositions can I rest satisfied that my problem is due to inadequate understanding of mathematics and not to a senseless " hyper-philosophising".

The intrusion of some form of metaphysics into science, eroding the notion of empirical validity traditionally at the core of the scientific method, becomes even more apparent today, when we increasingly hear theoretical physicists referring to entities ( see Higgs' bosons ) that are BELIEVED to exist, without a direct empirical observation, only on the basis of mathematically supported theoretical inferences or, at best, as dim “traces” inferred from the behaviour of other particles.

Ittiandro

[wtf]

@Ittiandro, Would your concerns be addressed if we agreed that scientific theories are descriptive rather than explanatory? In other words (as Newton pointed out), his theory of gravity only described how gravity works; the theory does not say what the underlying mechanism is. If we view science as descriptive, are your concerns satisfied?

[The Inglorious One]

“The final solution was approached in… turning around the question. Instead of asking: How can one in the
known mathematical scheme express a given experimental situation? the other question was put: Is it true, perhaps, that only such experimental situations can arise in nature as can be expressed in the mathematical formalism?” -- Werner Heisenberg

IOW, wtf is right in saying "scientific theories are descriptive rather than explanatory."

[surreptitious57]

So before there was biology there was chemistry
So before there was chemistry there was physics
Before there was physics there was mathematics
Before there was mathematics there was nothing
As it is not a human construct but something else

[ittiandro]

@Ittiandro, Would your concerns be addressed if we agreed that scientific theories are descriptive rather than explanatory? In other words (as Newton pointed out), his theory of gravity only described how gravity works; the theory does not say what the underlying mechanism is. If we view science as descriptive, are your concerns satisfied?

I am not too sure I understand what is the difference between providing a description for a given phenomenon ( e.g. gravity) and knowing its “ underlying mechanism”, unless by “underlying mechanism” we mean its first cause and/or its purpose ( a final cause ?) or we imply that phenomena, as they appear to us are not “ the things in themselves”. If this is what you imply, however philosophically relevant such speculations may be, they trespass the boundaries of the commonly accepted notion of science.

Newton’s “ theory” of gravity was certainly a “ description” of how gravity works, but, inasmuch as this description was based on empirically observable data and had ( as it still has today) predictive validity, it also provided an "explanation" of gravity , even more so that such an explanation was borne out very accurately by mathematics. To me, there is no contradiction in saying that all scientific hypotheses are descriptions and explanations at the same time, inasmuch as they are empirically verifiable, until a better explanation is found.

In the end,whether or not the distinction between description and explanation, when applied to scientific hypotheses, is valid, I still think it it does not address my initial question: even when our descriptions/explanations of reality are fully mathematically supported and have predictive value, my concern is to bring to light in our common language the link between the mathematical “encryption” and the corresponding CONCEPTUAL representation, underlying it .

Without this conceptual bridge available to us non mathematicians , it would be as if a cryptologist claimed to have found the key of an encrypted message, but then he failed to translate it into our common language, leaving to us, non specialists, the task to figure it out .

Sometimes, when faced with pages of mathematical equations and symbols purportedly containing the confirmation ( whether we call it description or an explanation) of a given hypothesis regarding the workings of nature i( be it gravity, curved space, etc), I dream to see a brief " conceptual" transliteration of each step of the mathematical sequence, allowing me to see the " conceptual" flow behind the conclusions rather than blindly accepting them... ..May be this, if feasible, would constitute a remarkable step forward towards the the common man's understanding of science..The question is : is this theoretically and practically feasible or does it perhaps run counter the very essence of mathematics? I think this is another interesting point to debate..

Ittiandro

[A_Seagull]

Physics models the real world. The mathematics describes the model. eg F=ma. There needs to be a mapping between the symbols of the mathematical model and the measurement of various physical entities, e.g. length, mass, momentum, energy.

The fact that the mathematical models can be used to make predictions about future measurements is an indication that the model is a good one. It says nothing about the nature of the underlying noumena or reality.

The actual form of the noumena is not of particular concern to physicists (although many of them might claim that it does). The actual form of the noumena is of concern to meta-physical philosophers for their speculation, (although they are unlikely to ever reach any meaningful conclusion.

Does this help?

[Hobbes' Choice]

Just to say, it would help if you broke up your post into paragraphs.

Welcome aboard.

PhilX

Attention spans are not what they used to be.

[Walker]

What is the nature of the understanding that physicists and cosmologists claim ( or strive ) to attain through their mathematical constructs, impervious, as they are, to conceptual representation?

The nature of the understanding is extrapolation, and math is a precise language.

Full Definition of EXTRAPOLATE
transitive verb
1: to infer (values of a variable in an unobserved interval) from values within an already observed interval
2a : to project, extend, or expand (known data or experience) into an area not known or experienced so as to arrive at a usually conjectural knowledge of the unknown area <extrapolates present trends to construct an image of the future>
b : to predict by projecting past experience or known data <extrapolate public sentiment on one issue from known public reaction on others>

*

Ultimately, I think the objective is to find the one statement that answers all questions.

[ittiandro]

Just to say, it would help if you broke up your post into paragraphs.

Welcome aboard.

PhilX

Attention spans are not what they used to be.

Thanks for your 2nd reminder, PhilX. I'm doing my best .

As you can see, I have already re-edited my original post by further breaking it down in paragraphs and I’ve done the same for my subsequent posts.

When dealing with intellectual and philosophical issues, however, there is a limit to how much one can break down the paragraphs without diluting or even breaking its logical thread.

Besides, this is a philosophy forum and I thought that readers would be more immune than others to… attention span problems, due to their familiarity with philosophical issues.

After all, who wrought havoc with the readers’ attention span more than Kant or Heidegger or the proponents of linguistic analysis, the latter with their lengthy essays on issues such as “ the cat is on the mat” sentence(!). Contrary to their alleged " clarifying" purpose, lucubrations of this type have always challenged my attention span, eliciting in me a strong urge to fall asleep or reach for... a Ritalin pill, if available.

Perhaps the fact that my question is unusual makes it difficult to understand and thus challenges even more the attention span . It may be even flawed in its presuppositions, but this is why I brought it up for a healthy discussion.

Cheers

Ittiandro

[wtf]

When dealing with intellectual and philosophical issues, however, there is a limit to how much one can break down the paragraphs without diluting or even breaking its logical thread.

"When writing about transcendental issues, be transcendentally clear." -- Descartes

Perhaps the fact that my question is unusual makes it difficult to understand and thus challenges even more the attention span . It may be even flawed in its presuppositions, but this is why I brought it up for a healthy discussion.

Two people have asked you whether your concern would be addressed by differentiating science as descriptive versus explanatory. You have not yet responded adequately IMO. To be fair, you did respond to me when I made that point; but not in a way I could understand. You didn't seem to think there's a difference between description and explanation; when in fact there is all the difference in the world.

[Obvious Leo]

Having just read through this thread it seems that most of the main points have already been covered but the question you raise is nevertheless a profoundly important one, ittiandro.

Mathematical physics is not designed to model the universe, although some of the media sluts in the physics business might have us believe otherwise. Mathematical physics is only able to model a particular narrative of the universe which must first be specified in advance. In other words instead of modelling what's going on in the universe physics models what the physicist thinks is going on in the universe. In this respect spacetime physics is no different from the Ptolemaic geocentric cosmology. It's fine for as long as it works and then it's just a crock of shit. Most of the most illustrious theoretical physicists in the modern era are well aware of the fact that spacetime physics pretty much went past its use-by date after Gell-Mann and the Standard Model.

The tautologous nature of mathematical physics is sadly not well understood by the legion of logical positivists who still linger in the camp. Physics is entirely based on observation and an observation is an act of cognition which is then formulated into a procedure of thought which we call a THEORY. The theory is then interrogated to yield predictions as to what the observer will observe but this tautologous methodology awards us only a Pyrrhic victory when the observer duly goes ahead and observes what the models have predicted.

“(...) Truth, it is said, consists in the agreement of cognition with its object. In consequence of this mere nominal definition, my cognition, to count as true, is supposed to agree with its object. Now I can compare the object with my cognition, however, only by cognising it. Hence my cognition is supposed to confirm itself, which is far short of being sufficient for truth. For since the object is outside me, the cognition in me, all I can ever pass judgement on is whether my cognition of the object agrees with my cognition of the object”
....Immanuel Kant....from the Jasche Lectures on Logic.

I'm not suggesting that physics can be done in any other way but it saddens me that so many of its most vocal practitioners seem to be unaware of this inherent limitation to their own discipline. In this sense physics is not a science at all because a true science must be both predictive and explanatory. Physics is more appropriately defined as a branch of applied mathematics because it meets only the first of these criteria.

[surreptitious57]

Physics is entirely based on observation and an observation is an act of cognition which is then formulated into a procedure of thought
which we call a THEORY. The theory is then interrogated to yield predictions as to what the observer will observe but this tautologous
methodology awards us only a Pyrrhic victory when the observer duly goes ahead and observes what the models have predicted

And yet that does not explain why quantum mechanics is so counter intuitive nor why it cannot be successfully unified with general relativity
If you are going to deny the validity of observation as a tool of science then all you are left with is logic and reason which are inadequate for explaining physical phenomena. For if evidence disagrees with a theory it has been falsified and a new one has to be discovered to replace it
And so theories are not there to establish confirmation bias of the scientific community as you mistakenly think. Science is predominantly an inductive discipline. So has to be eternally self correcting in order to allow for theories to be modifed when new evidence becomes available