Category Archives: Expansion of Universe

FLRW Metric and ‘Expansion of Space’

We start from Wikipedia’s explanation of ‘Metric Expansion of Space’:

“The metric expansion of space is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. It means that the early universe did not expand “into” anything and does not require space to exist “outside” the universe – instead space itself changed, carrying the early universe with it as it grew. This is a completely different kind of expansion than the expansions and explosions seen in daily life. It also seems to be a property of the entire universe as a whole rather than a phenomenon that applies just to one part of the universe or can be observed from “outside” it. Metric expansion is a key feature of Big Bang cosmology, is modeled mathematically with the Friedmann-Lemaître-Robertson-Walker metric (FLRW Metric) and is a generic property of the universe we inhabit. However, the model is valid only on large scales (roughly the scale of galaxy clusters and above), because gravitational attraction binds matter together strongly enough that metric expansion cannot be observed at this time, on a smaller scale.”

In short, according to official sources, ‘Expansion of Space’ stuff is rooted in FLRW metric. It is said that before the discovery of ‘redshift-distance’ relationship in light coming from far off galaxies in 1929 by Edwin Hubble, (F) Friedmann (1922) and (L) Lemaître (1927) already had described ‘Expansion of Space’ in their respective works.

Before presenting the actual points of (F) Friedmann (1922) and (L) Lemaître (1927), let me first share point of view of a famous Internet Physics writer Mr. Victor T. Toth on this topic. Following was his reply dated December 01, 2017 to a question:

Big Bang theorists do not claim that space is “created physically”, whatever that means.

Big Bang theorists do claim that things, on average, recede from each other; that the distance between things is therefore increasing, on average; and that correspondingly, the metric of spacetime evolves as governed by Einstein’s field equations.

None of this implies space being created, “physically” or otherwise. (For starters, space is not a measurable, tangible concept, nor is it a conserved physical quantity. When you measure “space”, what you actually measure is the distance between things, not space itself, which is intangible.)

In above answer, Mr. Victor T. Toth is saying that Big Bang Theorists do not claim that space is expanding or being created or anything like that. But in another recent answer, he is accepting that Big Bang Theorists do say like that and thus he is showing his disagreement with those Theorists:

Not for the first time, allow me to be the contrarian here and challenge my esteemed colleagues who are telling you that space is expanding, by making three (to me) rather important points: (i) What is this “space” that is expanding? How do you measure it? Where are its little markers to which you can attach your measuring tape? And exactly how is this “space” represented in the Friedmann equations? (ii) Speaking of which, if it was space expanding, how come I can derive (see, e.g., books by Weinberg or Mukhanov) the aforementioned Friedmann equations purely in the context of Newtonian physics, with its concept of absolute space and time? (iii) Last but not least, when gravity brings expansion to a halt, how does it do that? Is it somehow acting on “space”, as opposed to acting on matter? (See also Peacock’s Cosmological Physics.)

No, space is not expanding. It’s not even something we could measure if it did. The Friedmann equations contain two entities: matter (represented by its density and pressure) and the gravitational field (represented by one component of the very special, homogeneous and isotropic FLRW metric.)

Galaxies are moving further apart. If you could stretch a measuring tape from the Milky Way to a distant galaxy, the distant galaxy would be zipping alongside that measuring tape at quite a clip (probably several hundred kilometers a second, at the very least.) And when, in a region where matter is denser-than-average, gravity prevails, it stops those galaxies from moving away from one another.

The purpose of presenting quotes of Mr. Victor T. Toth was to show that some big bang cosmologists are already against the idea of Expansion of Space. However here Mr. Victor T. Toth is not representing the dominant opinion of mainstream big bang cosmologists who overwhelmingly think that Space is Expanding and that this notion of Expansion of Space is rooted in the works of (F) Friedmann (1922) and (L) Lemaître (1927).

Therefore, now I will show that both (F) Friedmann (1922) and (L) Lemaître (1927) did not actually talk anything about Expansion of Space and that this notion is deceptively being attributed to them by the mainstream cosmologists. Mr. Victor T. Toth already has given a hint that Friedmann equations contain two entities which are (i) matter and (ii) gravitational field and thus there is nothing like Expansion of Space in the works of Friedmann (1922).

So let us first check the Friedmann’s actual concept of space. The English Translated title of his 1922 paper is “On the Curvature of Space”. He uses terms ‘space’ synonymous to ‘radius of universe’. By the term ‘radius of universe’ his meaning is that mass contents of universe would cause gravitational boundary of universe that a straight line universal journey of a physical object would be a complete circle and would reach back to the original point. ‘Radius of universe’ is radius of this universal ‘straight’ line which is actually circular. Within this meaning of ‘space’, it is physically valid to say that space may expand or contract. Within mathematical model of Friedmann, space is really expanding or contracting according to this meaning. Following are some examples in Friedmann’s paper of usage of term Radius R as curvature of space:

“Here R depends only on x4 and it is proportional to the radius of curvature of space, which may therefore change with time.”

While deriving constant universe model of Einstein within his own general scheme,

Friedmann writes: “whereby R signifies the constant (independent of x4) radius of curvature of space.”

“If we restrict our consideration to positive radii of curvature”.

“Let the radius of curvature equal R0 for t = t0.”

“Positive or negative depending on whether the radius of curvature is increasing or decreasing for t = t0.”

“by choice of the time it can always be arranged such that the radius of curvature increases with increasing time at t = t0.”

It is now clear that yes space is contracting or expanding in Friedmann’s model but it is contracting or expanding within above physically valid meanings of contraction or expansion of space. But Big Bang Cosmologists tell us a whole different and misleading thing and they attribute their own faulty model to Friedmann. They call their own misleading model of ‘expansion of space’ as ‘metric expansion of space’ and wrongfully attribute this faulty physical model to Friedmann.

After checking the actual position of Friedmann (1922), now we come to see the actual position of Lemaître (1927) with regards to the notion of Expansion of Space.

Modern concept of Expansion of Space has actually come from manipulating Equation No.23 of Lemaître (1927) paper. Following is the snapshot of Equation No.23:

This equation can be written as V/C = (R’/R)r

The above form of equation No.23 superficially resembles to Hubble Law which is V = HD

In Equation No.23, V/C is ‘Redshift’ and in Hubble Law, V is ‘Redshift’; thus LHS of both equations are equal.

Moreover, in Equation No.23, r is Distance, so ‘r’ and ‘D’ of RHS of both equations are also equal.

Therefore, if we use the notation of Hubble Law, we can write Equation No.23 as following:

V = (R’/R)D

R means radius of whole Universe … (Radius of ‘whole’ universe itself should have been regarded as ‘cranky idea’ in first place).

Anyhow ‘R’ means radius of whole universe.

What Lemaître stated was like V=(R’/R)D

What standard ‘interpretation’ goes in every official source … books/papers etc. that is V=(S’/S)D

In short Lemaître was saying in his equation No.23 (1927) that redshift (V) is caused by increase of radius of whole universe. While distance of galaxy (D) remains constant.

Actual equation No.23 is not exact this one. If we use notation of Hubble law then equation No.23 becomes like this and superficially does resemble with Hubble law.

But unlike Hubble law where H is constant … here we have distance of galaxy (D) as constant.

R’/R … does it mean H or not?

Whether or not it mean H … it is not constant like H

This is the actual position of Lemaître .

What FLRW metric attributes to him?

FLRW metric makes this thing into V=(S’/S)D where S means ‘Space’.

Here conversion of R into S is a simple manipulation.

Lemaitre here did not say increase of Space or even increase of distance of galaxy… according to equation No.23, distance of galaxy remained the same.

This thing has been ‘interpreted’ in FLRW metric that ‘coordinate’ of galaxy remains the same and space is increasing.

In the end … after all this is a deceptive manipulation. V=(R’/R)D is NOT equal to V=(S’/S)D.

Thus we have seen and confirmed that both Friedmann (1922) and Lemaître (1927) had not coined the term or concept of Expansion of Space and that this concept or notion is only deceptively being attributed to both of them by the so-called FLRW metric.

Position is that without the notion of Expansion of Space, the Standard Model of Cosmology (Lambda-CDM) does not work and this notion itself is unreal, illogical, non-physical as well as deceptive.

For further details, please see my book “A Philosophical Rejection of The Big Bang Theory”.

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Cosmologists are finding new problems in their model yet they are not ready to accept that Universe is not Expanding

The latest story is that after finding that expansion rate of Universe is 9% faster than previously thought now they recalculate the age of Universe and say it must be one billion years younger than previous estimate of 13.8 billion years. The new estimate is around 12.5 billion years.

The universe may be a billion years younger than we thought. Scientists are scrambling to figure out why.

For me, these are not impressive results. These are pathetic. Researchers that include Nobel laureates are insisting that Universe is only 12.5 billion years old.

Universe is neither 12, 13 or 14 billion years old. Their model has inherent problems and they look into the reality only indirectly i.e. through their model. They are not ready to throw this model altogether. Still they are trying to find only minor mistakes in their model and somehow they will reconcile through fine tuning and simple adjustments. But fact is that Universe is not expanding at all and even the farthest visible galaxies may be located at distance of many hundred billion light years. I am going to explain this point in my second upcoming book titled “Philosophy Unscrambles Dark Matter”.

de-Sitter effect and the actual ‘prediction’ of redshift-distance relationship in year 1917:

Fritz Zwicky (so called father of dark matter) had proposed various alternative mechanisms to explain redshifts in those early days but he himself accepted that they did not work as expected.

Since all other ‘mechanisms’ failed … so only the ‘velocity’ mechanism survived.

But that was not just the survival of fittest thing.

Hubble was skeptical of velocity interpretation of redshifts but … actually he was also alone. He did not openly disregard velocity meanings. He cited de-Sitter in his 1929 paper that in the de-Sitter Cosmology, displacement of spectra arise from reasons other than motion.

But … de-Sitter also left him alone. Mainly because perhaps the specific point of de-Sitter was not having direct link with General Relativity.

After 1929, de-Sitter and Arthur Eddington both were anxious how to explain Hubble type redshift-distance relationship within GR framework.

Hubble was thinking that de-Sitter already had the prediction of cosmological redshifts in terms other than motion.

de-Sitter knew that he did have prediction but only by way of speculation or expectation out of observed data of few redshifted galaxies by time 1917.

de-Sitter did speculate that redshift could be systematically linked with distance and he had speculate this thing in year 1917.

http://www.pitt.edu/~jdnorton/te…

Above is de-Sitter (1917) paper. The speculation about redshift-distance can be seen at the end of page No.26.

He is saying that distance may cause redshifts , giving rise to spurious positive radial velocity.

Meanings of ‘Spurious’ can be seen here:

SPURIOUS | meaning in the Cambridge English Dictionary

false and not what it appears to be, or (of reasons and judgments)based on something that has not been correctlyunderstoodand thereforefalse:

On the next page (27), he is saying that Helium stars actually show this type of displacement of (spectra) … On same page, he is using term ‘apparent velocities’ for galactic redshifts.

Who was Willem de Sitter – Wikipedia?

After development of GR equation in 1916 … Einstein was perhaps the first who developed a model of whole universe based on GR equations in year 1917.

de-Sitter was the second person who also developed a model of whole Universe based on GR equations in same year 1917.

The 1917 de-Sitter paper is like a ‘solution’ to GR equations or at least regarded as solution to GR equations.

In this 1917 paper, de-Sitter is saying that distance may be linked to ‘spurious’ velocity (redshift).

In this paper … he did develop solution to GR equations … but there is lot of general discussion as well … like various developments in astronomy etc. including redshifts.

Now whether had ‘Spurious Velocity’ come from GR or from general speculation?

If it came from GR, then why de-Sitter and Eddington … after 1929 … were sitting together to find a solution of Hubble type redshifts from within GR equations and were not reaching at satisfactory conclusion…???

I do not know answer to this question. Only justification is that de-Sitter knew that page 26–28 of his 1917 paper were mere wild speculations and thus they did not relate to the current problem of how to account for Hubble type redshift-distance relationship within the framework of GR.

de-Sitter and Arthur Eddington were actually worrying in year 1930 regarding how to account for Hubble type redshift-distance relationship within the framework of GR equations. Following reference tells the story that they were worrying on this point:

http://www.physics.umd.edu/grt/t…

The story can be found on page 8 of this PDF file (PDF page).

The rest of the story is that Lemaitre approached Eddington that he already had published similar solution in year 1927.

Then Eddington got a plan. He advised Lemaitre to publish the translation but with modifications.

The English translation (1931) of Lemaitre paper changed the game. That was a modified and deceptive translation … showing as if Lemaitre had already (in 1927) derived Hubble Law from GR equations.

In this way, Hubble was left alone. de-Sitter also adopted Lemaitre explanation. Zwicky proposed alternatives but then withdrew his proposals not because he was satisfied with expansionism … only because he had not found viable alternative.

About Latest developments:

It is often stated that latest developments have confirmed expansion of universe. Actually there is no latest development with regards to providing direct evidence that galaxies are in fact in motion. Even the so called ‘inflation theory’ has been ‘derived’ from de-Sitter model where cause of redshifts was other than motion.

Unscientific methodology of Science within the Big Bang Cosmology

Modern science, especially the Big Bang Cosmology, tends to explain little (observable) facts on the basis of ‘already known’ realities of whole universe. For example, our scientist (Georges Lemaître) already knows that ‘Universe is Expanding’. On the basis of this ‘already known’ larger fact of reality, he ‘explains’ observed redshifts of galaxies (little fact).

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Title of his (1927) paper is “A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra-galactic Nebulæ”.

Thus before explaining redshifts, he is already aware of ‘larger fact’ that radius of universe is increasing. On the basis of already known ‘larger fact’, he is explaining or interpreting a little (observable) fact.

CMBR is also explained in this way. Science ‘already knows’ that universe originated from a point that, after 380000 years, had become 43 million light years in diameter. Science ‘already knows’ that this universe first time emitted light (photons). Science ‘already knows’ that that universe had such and such temperature. Science ‘already knows’ that universe was expanding and that light was also expanding. Science ‘already knows’ that now the original light must have this much temperature.

On the basis of these ‘already known’ larger realities of universe, CMBR (a little observable fact) is explained.

‘Predictions’ of Big Bang Cosmology did not even match with observed fact. Therefore, adjustments of dark matter/dark energy are applied to get matching results. This is the actual prevailing scientific methodology whose examples are quite apparent in different aspects of the Big Bang Cosmology. And they say that it is science.

Can Mathematics discover unknown realities of Physical World?

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.

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.”