Mathematics is quantitative extension of logic. It can move beyond logic only in exact quantitative terms. The basic realities are discovered through observations. Secondary realities are discovered through logic. Quantitative realities are derived from basic and secondary realities by using mathematics.
Unfortunately Modern Physics is built on assumption that mathematics itself is capable to discover unknown realities of Physics. We often read in science literature that physical fact ‘x’ was already ‘predicted’ by equations ‘M’ so physical fact ‘x’ is considered as a necessary consequence of equations ‘M’ and interpreted only within the context of same equations ‘M’.
In the Big Bang Theory, there is a deceptive claim that in year 1927, Georges Lemaître had ‘mathematically predicted’ physical fact of linear relationship of redshifts and distance whereas that mathematical prediction came true in year 1929 when Edwin Hubble experimentally found the same linear relationship. Only because the said relationship was already mathematically ‘predicted’, mainstream Physics did not feel the need to see whether that kind of relationship could be considered as physical proof of expansion or not. Only because already available mathematics was talking about ‘expansion’ so the newly found linear relationship was simply interpreted in terms of ‘expansion’.
Likewise, CMBR type radiation were already ‘mathematically’ predicted. When apparently same type of radiation were experimentally found, then again mainstream Physics felt no need to find the actual reason of those radiations and they were simply interpreted within the context of already available mathematics.
Right method should be Observation>>>Logical Interpretation of observed reality by evaluating all the possible explanations>>>Quantified Model using Mathematics. (This method implies that mathematics itself does not find unknown realities)
The actual prevailing method of Modern Physics is like Mathematics>>>apparent resemblance of later on found physical facts with already available mathematics>>>Hue and Cry that ‘mathematical prediction’ has come true>>>adaptation of already available mathematics (with few modifications) as final interpretation of newly found physical facts. (This method implies that only way to find unknown is the way of mathematics)
It is often stated that Einstein had found hard physical realities through the way of mathematics. Bending of light ray was experimentally confirmed during a Solar Eclipse etc. The thing to be noted in this case is that after reaching at ‘Equivalence Principle’ with the support of Eötvös experiment (Observed physical fact), First of all Einstein had logically evaluated the stuff. If acceleration due to gravity was independent of mass (a logical interpretation of physical experiment) then light also must accelerate under gravity (logic). But according to his own theory, speed of light could not be affected (logic). If speed of light could not be affected then there were two (logical) options; (i) Wavelength of light could be affected (gravitational redshifting) or (ii) light could change direction (i.e. other logical form of acceleration).
Thus logic took him to the idea of bending of light. Next stage was quantification i.e. ‘how much bending’. From this point onward the role of mathematics entered to the scene.
Therefore mathematics had not found unknown reality in this case. More specifically, if Solar Eclipse experiment had really confirmed anything then it was that (logical result out of observation) acceleration due to gravity was independent of mass.
However if we try to look at further details, then keeping in view the personality of Arthur Eddington who actually performed 1919 Solar Eclipse Experiment, the whole confirmation of General Relativity on the basis of this experiment sounds notorious and unreliable.
Arthur Eddington, within mainstream interpretation, had confirmed that mathematics really found unknown realities. Eddington again played a crucial role in 1931 when it was almost established that Georges Lemaître had already ‘mathematically predicted’ Hubble’s law in year 1927. However, later on, the same Arthur Eddington, in the case of Subrahmanyan Chandrasekhar, would refuse that mathematics can find unknown realities of Physics.
But since Chandrasekhar’s point of view was succeeded at the end so mainstream Physics ignored the confession of Arthur Eddington that Mathematics itself cannot find realities of Physics.
Role of mathematics is also important to be clearly explained and limits be identified within the domain of Philosophy. Below I am quoting a relevant portion of my upcoming book “Descriptive Knowledge, Mind and Reality; a case of Epistemological Realism”:
“There is no a priori knowledge which is totally independent of sense experience and neither mind is a flat recipient of sensory information. Those general tendencies form a natural flow of expression of contents of framework of consciousness and tend to make it consistent, smoother, balanced and/ or more accurate. This is logical way of how external reality is perceived and then expressed and does not amount to a priori knowledge. To say that mind derives things or can calculate is equivalent to accept that there is no a priori knowledge. Mind derives or calculates means that mind has the ability to derive or calculate and does not mean that mind is already aware of correct answers. Certainty that we get from the results of mathematics is not real as most of the times it depends on suppositions. One plus one is always two because quantity of one is supposed to be fixed. Mathematics in its practical usage by mind is based on suppositions and thus not real; however there comes a real aspect of mind that mind is able to suppose fixed, unchanging, absolute or universal entities and then becomes able to get certain results by performing mathematical or logical operations on those supposed universals. Universals themselves are unreal but ability of mind to suppose them and perform mathematics and logic on them is real. Universals come from ability of mind to suppose and not from ability of mind to generalize. Generalization leads to ultimate categorization and not universalization. Simple analytic judgments are also not a priori. Judgment, basically being one or the other form of inference or conclusion, itself is a secondary thing. At the most it is a mold which gives the primary sensory information a different and useful shape such that all the ingredients of final product were already contained in that primary sensory information. In this capacity, again it is ability and not pre-existent correct answer. In analytic judgments where predicate is obtained by simply analyzing the subject, the pre-existent correct answer was contained in primary sense data and not created by mind through judgment. We take example of analytic judgment as provided by Lord Bertrand Russell[i] which is stated as: “a bald man is a man”. This type of analytic judgment was regarded in pre-Kant era, Bertrand Russell states, as example of a priori knowledge because in these judgments, predicate being part of the subject, we are certain a priori. As already mentioned, judgments themselves are secondary in origin therefore certainty connected with them is also secondary in character thus there is nothing a priori in analytic judgment. This is equivalent to say that it is certain that a tree is tree so knowledge of tree is a priori or at least this judgment is a priori. Knowledge of tree comes from senses and judgment is a secondary thing which tells us nothing wholly independent of sensory information.
Issue of analytic judgments, as Russell continues, is connected with principle of contradiction which asserts that nothing can at the same time have and not have a certain property. We should now examine the case whether this principle itself can be regarded as innate or a priori or not. The certain thing is that this principle in descriptive form is not innate. Even in the capacity of a logical mold machine, it is doubtful that this principle is innate of mind. However it is a fact that mind is not able to form mind images of self-contradictory things like impossible figures. But how an impossible thing can be regarded as innate just because mind remains unable to perform that impossible task? If principle of contradiction is innate of mind then inability of mind to see view of sun which is less than 8 minutes old is also innate of mind. The principle of contradiction is a general characteristic of outer reality which does not accommodate any object that at the same time has and not has a certain property. If we accept that this principle is innate of mind then next claim that outer reality is constructed by mind also becomes possible because then outer reality would be obeying this principle of mind and could well be a product of mind. In this regard, what is innate of mind is the ability of mind to remain consistent with outer reality though mind is able to stay away from facts of outer reality also. Basically human mind tends to stay away from facts and only after sufferings through mistakes then becomes able to keep itself consistent with outer reality. With the passage of this evolutionary process, mind has then captured this principle from the general behavior of outer reality through sense experience and through its own inability to form mind image of self-contradictory things. Then mind has given it absolute form through innate ability of supposition of universals. This principle talks of universal, unchanging and fixed entities; such things being non-extractable from general behavior of outer reality so this ability of supposition is innate of mind. But what things can be assigned universal attributes? The answer is only those things whose all component parts are traceable to sense experience, details thereof we shall see in chapter 5. Here we can proceed with our conclusion regarding analytic judgments and principle of contradiction that both are not innate and do not amount to a priori knowledge. However with regard to principle of contradiction, we do acknowledge that ‘inability’ of mind to imagine self-contradictory idea in the form of mind image is innate but at the same time this inability does not constitute any innate or a priori knowledge of mind which is independent of sense experience. Examples include impossible figures which cannot be imagined in the form of mind image and neither can be drawn on paper except in the form of illusory tricks. But this inability of mind is consistent with the inability of outer reality where also any impossible figure cannot exist. Self-contradictory thing is an impossible thing in itself and does not depend on inability of mind for its impossibility of physical existence. In simple terms, mind is not able to imagine what is impossible in itself so there is no involvement of any positive ability of mind. Positive ability of mind in this connection is the ability to conceive self-contradictory things only in descriptive format. It is possible for mind to conceive descriptive concept of impossible figure like a ‘square circle’ but it is not possible for mind to form mind image of same concept. Mere ability or inability of this sort does not constitute a priori or innate knowledge of mind. Our conclusion regarding ability of supposition of universals is that this ability is innate to human mind only since this ability seems not available to animal mind and that things that are supposed as universals ultimately had come from sense experience and thus there is no involvement of a priori knowledge.
If we take simple meaning of a priori knowledge as something we do not need to confirm in our routine judgments then yes we know a priori that a bald man is a man. Or that 7+5=12 is also a priori knowledge. But through this philosophy, we are examining our basic tools and infrastructure. In this realm, innately we are not even provided with any idea of 7 or 5. Children of pre-school age can learn counting only if they are taught by parents or guardians. Few months back I met a 10-11 years old boy who does not know counting more than 20 because his father died when he was so young and due to difficult domestic economic conditions, he might have attended school for only a short duration. Suppose he can do 2+2=4, which he cannot do in fact but suppose he can do, but how he will do 22+22=44? Suppose up to 20 figures are innate in him but why the simple figure 103 is not innate in him? It was a slip number which he could not read. My 4 years son who is studying in prep class can count up to 100 but he cannot do simple sum of 2+2=4 without my help and guidance that I provide using examples of concrete things. Lord Russell further states and affirms with reference to works of David Hume that 7+5=12 is a synthetic judgment as idea of 12 is not contained in 7 or 5; not even in idea of adding them together and also that knowledge of pure mathematics is a priori. Russell seems to point out that with this new problem, synthetic judgments also became a candidate of a priori knowledge. In fact David Hume has not discussed analytic or synthetic stuff at least in Enquiry Concerning Human Understanding. And in my humble opinion, David Hume has taken pure mathematics as a priori only in routine sense which I have described at the start of this paragraph. After discussing mathematics as a priori, then Hume moves to matters of fact such as someone is in Paris or not etc. that needs factual evidence for knowing the truth. Then he discusses cause and effect and concludes that cause and effect cannot be known a priori. His following sentence will make sense that basically he is talking about a priori knowledge in routine sense.
“I venture to assert, as true without exception, that knowledge about causes is never acquired through a priori reasoning, and always comes from our experience of finding that particular objects are constantly associated with one other.”[ii]
While discussing mathematics, his point was that reasoning alone could perform arithmetic or geometric operations and experience was not required. With the problem of cause and effects, his opinion is that reasoning alone is insufficient and experience is required. With further proceedings, more troubles will arise and he will say that past experience will also fail to predict future occurrences. Anyways, I here disagree that only pure reasoning can solve arithmetic or geometric problems as no pure reasoning which is independent of sensory experience exists at all because reasoning is the name of inferring process which is not independent of sense experience. However by saying that through reasoning, problems of arithmetic etc. will be solved, he is clearly taking routine meanings of a priori knowledge and same is the case with his later analysis. And perhaps Kant also has example of only mathematics with regard to synthetic form of a priori knowledge which we accept a priori only in routine sense and not in the sense of basic innate a priori knowledge of human mind. On the whole yes we are a priori certain that 7+5=12 but what about mathematical assertion that sum of series of odd numbers like “1+3+5+…” is always a perfect square? Only a person who already knows it through experience would be a priori certain about it. Any person who has not yet experienced it would tend to confirm it by adding 1 with 3 to get 4 which is a perfect square and then would add 5 with earlier answer 4 to get 9 which again is a perfect square and so on up to a reasonable limit to get a priority surety for the next times. David Hume had rightly pointed out (i.e. with only adjustment that pure reasoning is not independent of experience) that by examining only reasonable instances of mathematical assertions, we can get a priority surety for the future with regard to all the rest of un-examined infinite instances. But this is not the case with cause-effect situations where a priority surety for the future occurrences is not possible regardless of how many instances we already have examined.”