Is space really expanding? In a book called "Einstein, Relativity and Absolute Simultaneity" there was this sentence by Smith:

There is no observational evidence for a space expansion hypothesis. What is observed are superclusters of clusters of galaxies receding from each other with a velocity that is proportional to its distance.

He goes on to say space is Euclidean and infinite. Wouldn't this mean Big Bang was a explosion in spacetime rather than a expansion of spacetime as it is often told?
Is Smith just wrong or don't we know yet?
 A: There is observational evidence for space expansion, called redshift:  Most galaxies are moving away from us and we can measure that by the amount they are redshifted.  There are more details here.  But anyway, this is exactly what the author seems to deny as evidence.
Hubble noted that distant galaxies had higher redshifts and fitted his observational data with the famous linear relation now known as Hubble's law:
$$z=\frac{H_0}{c}r,$$
where z is the redshift, c is the speed of light, r is the distance and $H_0$ is Hubble's constant.  The further away a galaxy is, the faster is receding from us.
The question is, why are galaxies receding from us according to Hubble's law?  How do we interpret this equation?
There used to be the "tired light hypothesis", which tried to explain the Hubble relation by assuming that the galaxies are not receding from us, but that photons simply lose energy as they move through space.  This idea quickly fell out of fashion.
Another hypothesis is that the universe is expanding, which is exactly what we count as a fact nowadays.
The key point is that galaxies are not just receding from us, but also receding from each other.  We know that from the Cosmological Principle, which states that the universe is homogeneous and isotropic.  We do not live in a special place in the universe, but instead the universe is pretty much the same everywhere.  There are tons of evidence supporting the Cosmological Principle, and most of established cosmology is based upon its validity.
Philosophically speaking, we do not know that the universe is expanding, as we also do not know that the Cosmological Principle is valid.  If you analyze any scientific evidence philosophically, nothing would count as evidence.  Yes, galaxies could be just receding from each other due to some random cause, or maybe due to God, but the scientific evidence is clear: The universe is expanding and this is supported by both General Relativity (the Friedmann equations) and observational data (Redshift).  What we see is exactly what you would expect from a universe that is undergoing a homogeneous and isotropic expansion.  If you reject that the universe is expanding, or that the cosmological principle is valid, you may as well reject all of Cosmology.
A: 
There is no observational evidence for a space expansion hypothesis. What is observed are superclusters of clusters of galaxies receding from each other with a velocity that is proportional to its distance.

This is essentially correct. The expansion of the universe is just the recession of superclusters of galaxies. There are a lot of people who think there is some additional phenomenon of "expansion of space itself" going on, but there isn't. There isn't any concept in general relativity of "expanding space" (or "space", for that matter), and when it appears that there is, it's because you're getting confused by coordinate systems. See this answer.

He goes on to say space is Euclidean and infinite.

I suppose this is technically correct in the current standard cosmological model. Slices of constant cosmological time are assumed to be Euclidean and infinite in the model, and it fits the data so far. But the universe isn't Newtonian or special relativistic, because spacetime is curved in the model even though the spatial slices aren't.

Wouldn't this mean Big Bang was a explosion in spacetime rather than a expansion of spacetime as it is often told?

It doesn't imply that, no. It is possible to embed the cosmos that we see in some larger spacetime, and in some sense it would then be "expanding in" that larger spacetime. But there's no evidence to support any particular model of that kind. There doesn't need to be anything outside for the universe to expand into, and in the standard cosmological model, there isn't.
A: According to its Introduction, Einstein, Relativity and Absolute Simultaneity is a volume of essays “devoted, for the most part, to arguing that simultaneity is absolute” (as the title suggests). This is not mainstream physics. Since the book’s editors (William Lane Craig and Quentin Smith) are/were philosophers rather than physicists, its value as a cosmology textbook is doubtful.
So, yes, according to the weight of the available evidence and the consensus of mainstream physics, Smith is wrong.
A: Smith is just wrong.
He is right to say that there is no direct  observational evidence (in a naive sense) that space is expanding rather than space being fixed while galaxies spread apart through it. They both just look like galaxies getting farther apart.
But there is a lot of observational evidence that special and general relativity are correct. And, if those theories are correct, the only way to make sense of the observations is to say that space itself is expanding.
It's kind of like standing in a highway and seeing a car getting bigger and bigger and saying that there is no evidence it's getting closer because maybe it's just actually growing in size. We know cars just don't do that.
The problem is that, at small scales and velocities, the model Smith wants to support really works well, unlike the "growing car" model. So it's not obviously dumb to think that the galaxies are just flying apart through unchanging space. But, when you understand how things work at near-light speeds, it becomes clear that it just can't be right.
A: He's wrong.
What is observed, is a wide range of phenomena, not just one, and they're all pointing in the same direction.
Mainstream physicists aren't naive. These kinds of theories get tested hard against the data in a hundred ways and experiments, and against other theories, then all over again by other teams. They aren't just bought into because they are pretty and esoteric. As well as distant objects moving away faster:

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*objects moving away, seem to move similarly away in any direction. There is no apparent "centre". Either we are in some magical location, or it looks that way from everywhere... which would strongly imply space itself is expanding.

*same for structure of the universe (galaxies and voids), gamma bursts, supernovae, quasars.... Similar in all directions, no apparent "centre"

*cosmic microwave background has cooled over time. These are free-flying photons from the start of the universe. Photons just don’t "cool" as such. Not that way, and especially not gradually. There is no evidence that photons behave differently over cosmic times and distances (or rather when tested these ideas don’t win out). But they do gradually cool if space expands during their billions-years journey. This is direct evidence, that directly matches other redshift evidence.

*other experiments show that space is exceedingly "flat". But it didn't start that way. Again this is good evidence space has expanded. Like inflating a balloon with an ant standing on it, it'll look to the ant, very wrinkled when small, very much flatter when expanded. Our universe appears to us, to be extremely flat.

*certain events in the very early universe, such as reheating and supercooling, make sense and fit in mathematically, in a context of expansion, more than any other known suggested context that has been studied.

*Expansion is quite strongly suggested by Einsteins Field Equations, to the point that Einstein, who didnt have our current knowledge of the evidence, had to "fudge" his equations to have a way that they wouldn't end up suggesting that space was expanding or contracting. Einsteins field equations have been massively tested by everything from GPS to light deflection around stars, to black holes, to gravitational waves, to.... well, huge numbers of things. They describe how space itself behaves, primarily but not only in relation to gravity and mass. We are very sure of them so far. And if you dont modify them by throwing in a precalculated "cosmological constant" designed to deliberately counter the "otherwise normal" behaviour of the equations, they likewise show that space expands or contracts. Einstein didnt like doing that, but apparently couldnt see any other way to avoid that conclusion from his equations... people at the time believed in a static universe,and if that was so, then a "fudge factor" had to be added into his equations to make them predict what was mainstream believed, and not predict expansion.(These days, the "cosmological constant" is a little better understood and is now believed to relate to so-called "dark energy" - essentially one or more scalar quantum fields believed to exist and to be responsible for this expansion. The term "dark" means we don't yet have instruments that can directly perceive its energy or particles: it's "dark" to existing instruments,although we believe it's there from observations of the effects which match what such a field would do. We think about 69% of the entire mass-energy of the universe is in the form of this dark energy, so it's quite a big item cosmologically!)

*These theories have also made detailed and extremely accurate predictions, which are very hard to explain in other ways. For example, the Big Bang model explains why there should be a cosmic background of photons, and (if we figure a way to detect it) a cosmic background of neutrinos too, and some of their expected properties. It predicts the universe should contain 75% hydrogen, 25% helium, and traces of other elements, and exactly why those percentages (answer: it relates to the ratio of protons and neutrons created earlier on, and the time available for fusion), why some deuterium could form by fusion but then didn't fuse to helium (it fuses to helium very easily), and so on. No other known theory accounts for all such findings. And the Big Bang model includes as a core element, expansion.

Those are examples. There are others too.

*

*Relevant link for a bit more info:https://en.wikipedia.org/wiki/Expansion_of_the_universe
A: There is a lot of prose about the "stretching of space", similar to an elastic fabric. I would rather just say: things are moving apart. In technical language, I refer to the expansion tensor $\Theta_{\mu\nu}$ (one part of the "kinematic decomposition") for worldlines of galaxies, within a Friedmann-Lemaitre-Robertson-Walker model of the universe. At each point, this quantity tells you the relative motion of nearby galaxies. Emphasis on this quantity avoids unjustified poetic interpretations, which no doubt motivated Peacock's "Diatribe".
Another reason, I think, is reference frames. In Newtonian physics, you can take a single reference frame for the entire universe. Suppose it were centred on the Milky Way. Then, distant galaxies would recede at $2\times$, $3\times$, $4\times$ etc. the speed of light, relative to this frame. However in relativity, reference frames are best treated as local. Nothing goes faster than light in any (local, timelike) reference frame. Hence Davis & Lineweaver (2004) conclude "the velocity is due to the rate of expansion of space, not movement through space".
Some comments suggested physicists are better qualified than philosophers to address this topic. I disagree. It concerns interpretation of physics, and so philosophers of physics would in principle be the most authoritative. (Though in practice, to my knowledge it is physicists who have authored the relevant papers, including those cited above; also Luke Barnes and Geraint Lewis and collaborators; George Ellis, etc.) To be more nuanced, this is an intersection of different fields. What redshifts do galaxies have? Let's ask the astrophysicists, telescope technicians, statisticians etc. Which equations does general relativity produce? Let's ask the relativists. What is the meaning and interpretation? This is more the philosophy domain. (Of course, I do not imply everyone must stick to their own domain.)
A: He is basically wrong. What exactly is receding from what depends on the coordinate system you're using. In GR, all frames are physically equivalent to each other.
In comoving coordinates, ie. the Friedmann-Lemaitre-Robertson-Walker metric, the Universe consists of a uniformly expanding gas of galaxies. In this case, the galaxies move on geodesics associated to this metric. However, in comoving coordinates the galaxies have a fixed position and their velocity is $0$. So what does changes: the metric. The spatial part of the metric has a scale factor which changes with time. This is then associated to "the expansion of space."
We can also describe the same physics using local Minkowski frames. Since the Minkowski frame is static, we do not have an expansion of the spatial part. However, the cosmological redshift can now be attributed to the accumulation of Doppler shifts of photons sent by one galaxy and received by another through local Minkowski frames. In this frame, the velocity of the galaxies are not $0$ and they recede from each other.
To recap, it's a matter of perspective to decide what is "expanding." The first observational evidence is attributed to Edwin Hubble.
On the subject of the Big Bang, it's commonly believed that there was no spacetime before the Big Bang happened. There is not where or when, because there was no spacetime. The Big Bang happened and along came spacetime.
Of course, there are other theories like the cyclic universe.
A: Three concrete observational pieces of evidence can be linked to the transport of light/photons through the vast space of the universe. These are just examples -- not the body of evidence.

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*If you accept the big bang and the production of a 3000 K blackbody radiation about 300000 yrs after the big bang, where are those photons today? If the "explosion would happen into the spacetime" the shape of the spectrum would still be the same today, the photons would only be diluted. If "spacetime is exploding" the spectrum would be deformed by changing the wavelength (-> size) of the photons. The current CMB blackbody radiation of ~2.7 K is exactly what you expect in this case.


*Just think about: What does it mean, if astronomers say "an object has a redshift of X"? The redshift is seen in the pattern of very narrow/precise emission and absorption bands of normal molecules and atoms, which are at a very very different wavelengths (much longer) than they would be observed in a lab at the earth, the further an object is distant to earth. If the "big bang would be an explosion into spacetime" a photon would never change its wavelength, it would only be diluted or absorbed (intensity). Only in an "explosion of spacetime" the photon (wavelength) itself would be stretched, too, while it moves and its wavelength would become longer, the longer it travels through space. That is what we see.


*If you observe the light of extremely distant objects: on its way to us (as observers) it passes occasionally through areas of enhanced hydrogen (matter) density that will leave a typical and well known imprint of neutral hydrogen absorption bands. These bands are redshifted according to the distance of the (dark) hydrogen cloud. Since for very distant objects this will happen many times, we observe a series of many such shifted absorption bands. This is called the 'Lyman alpha forest' (https://en.wikipedia.org/wiki/Lyman-alpha_forest) and is a great tool to study the structure of the universe. This can be very well understood in the context of expanding space.
It was commented in the discussion below that those effects may just be kinematics (Doppler effect). While kinematics is never irrelevant, and moreover dominates our local environment on Earth and also in the local universe, on cosmological scales it becomes a minor correction. Redshifts we observe would correspond to highly- or ultra- relativistic speeds. And the observed isotropy of space is very consistent with the expansion of space, while it is not natural in a scenario driven by kinematics only.
