Logical positivism is a philosophy of science that was popular in the 1930s, and largely abandoned in the following decades. It is a variant of positivism, a view that is peculiar because it is widely accepted among scientists today, but widely rejected among philosophers of science. Wikipedia says logical positivism is "dead, or as dead as a philosophical movement ever becomes". [1,2]
I defend logical positivism, based on my understanding of it, as the best way to understand math and science today.
The core of logical positivism is to classify statements based on how we can determine their truth or falsity. There are three types.
1. Math. This includes mathematical theorems, logic, tautologies, and statements that are true or false by definition.
2. Science. This includes observations, measurements, induction, approximations, hypothesis testing, and other empirical work.
3. All else - mysticism, ethics, subjective beauty, opinion, religion, metaphysics, aesthetics, etc.
Mathematical truth is the highest kind of truth, and is given by logical proofs. It goes back to Euclid and the ancient Greeks, and was not independently discovered by anyone else, as far as I know. When a theorem is proved, you can be 100% sure of it.
You might think that mathematical knowledge is empirical, but it has not been for two millennia. You might also have been told Goedel's work somehow undermined the axiomatic method. The truth is more nearly the opposite. Math means logical proof. If I say that the square root of two is irrational, that means that I have a proof.
Science arrives at somewhat less certain truths using empirical methods. A conclusion might be that the Sun will rise in the East tomorrow morning. This is backed up by many observations, as well as theory that has been tested in many independent ways. Science also makes predictions with lesser confidence. These are considered truths, even though they are not as certain as mathematical truths.
There are many other statements that are not amenable to mathematical proof or empirical verification. For example, I may be of the opinion that sunsets are beautiful. But I do not have any way of demonstrating it, either mathematically or empirically. If you tell me that you disagree, I would be indifferent because there is no true/false meaning attached to the statement. It is meaningless, in that sense.
So far, this is just a definition, and not in serious dispute. What makes logical positivists controversial is their strong emphasis on truth. They tend to be skeptical about aspects of scientific theories that have not been tested and verified, and they may even be contemptuous of metaphysical beliefs. Others don't like it when their opinions are called meaningless.
Logical positivism is part of the broader philosophy of positivism that emphasizes empirically verifying truths. Logical positivism is a kind of positivism that recognizes math as non-empirical truth. When speaking of science, positivism and logical positivism are essentially the same. 
Logical positivists do distinguish between logical and empirical truths. I. Kant previously drew a distinction between analytic and synthetic statements, with analytic being like logical, and synthetic being like empirical. However he classified 7+5=12 as synthetic, which makes no sense to me. So his views have little to do with logical positivism.
Some statements might be a mixture of the three types, and not so easily classified. To fully understand a statement, you have to define its terms and context, and agree on what it means for the statement to be true. For the most part, mathematical truths are the statements found in math textbooks and journals, and scientific truths are the ones found in science textbooks and journals.
Enabled by XX century progress
I might not have been a positivist before the XX century, when broad progress in many fields provided a much sharper view of the known world.
In math, the major foundational issues got settled, with the work of Goedel, Bourbaki, and many others. Nearly all mathematics, if not all, has proofs that could be formalized in some axiom system like ZFC. 
This opinion may surprise you, as there is a common philosophical view that the pioneering work on logicism by G. Frege, B. Russell, and others was disproved by Goedel. One encyclopedia says: "On the whole the attempt to reduce mathematics to logic was not successful."  Scientific American published a 1993 article on "The Death of Proof". Not many mathematicians would agree.
The math journals publish theorems with proofs. The proofs are not written in a formal language, but they build on an axiomatic foundation and could be formalizable. It is not possible to have a Turing machine to determine the truth or falsity of any mathematical statement, so math is not reducible to logic in that sense. But every math theorem is also a logical theorem of axiomatic set theory, so in that sense, math has been reduced to logic.
In physics, advances like relativity and quantum mechanics expanded the scope of science from the sub-atomic to the galaxy cluster. Also these theories went fully causal, so that physics could hope to give a complete explanation of the workings of nature. The fundamental forces have quantitative theories that are as accurate as we can measure.
Again, this opinion may surprise you, as philosophers commonly deny that fundamental physics has anything to say about causality.  If that were true, it would rob physics of nearly all of its explanatory and predictive power. In my opinion, incorporating causality into fundamental physics was the most important intellectual achievement in modern times. Before 1850, we did not have causal explanations for any of the four fundamental forces. Now we have fully causal theories for all four, and those theories have led to the most striking technological progress in the history of the planet.
In biology, the discovery and deciphering of DNA and the genetic code meant that life could be included in the grand reductionist scientific vision.
Medicine, statistics, economics, and other fields made so much progress that XX century knowledge is the only thing to study.
With all the advances of the last century there is hardly any need to believe in unproven ideas, as ancient people might have attributed storms to the gods being angry. Today one can stick with positive knowledge, and have a coherent and satisfactory view of how the world works.
Philosophers have turned against positivism, and especially logical positivism. They consider the subject dead. The most common criticism is to attack metaphysical beliefs that logical positivists supposedly have. For example, some say that logical positivists cannot prove the merits of their views.
This is a bit like criticizing atheists for being unable to prove that there is no god.
Logical positivists do not accept metaphysics as truth. No argument about metaphysics can possibly refute logical positivism.
Some say that Goedel disproved the logical part of logical positivism. That is not true, as explained above. Goedel gave a clearer view of the axiomatic method, but did not undermine it.
The argument that logical positivism is self-defeating is fallacious in the same way as the argument that Goedel's logic is self-defeating. It misses the point about what the subject matter is. It is like saying: The imaginary numbers prove that the real numbers are incomplete, and therefor real analysis is invalid. No, the imaginary numbers mere shed light on what non-real numbers might look like.
Another line of criticism comes from what I call negativism. In this view, there is no such thing as positive truth. There are only falsified theories, and theories that have not yet been falsified. In effect, they say you can only prove a negative, not a positive. 
This seems backwards to me. If Newtonian theory is good enough to send a man to the Moon, then obviously there is something right about it. It is a true theory, and valid where it applies. It is not wrong just because it is not perfectly suited to all tasks. Pick up any science journal, and you will find it filled with positive knowledge.
Some science historians claim that the progress of science was fueled by anti-positivists who were willing to stake out beliefs that went beyond what could be empirically demonstrated. For example, some positivists were skeptical about the existence of atoms, long after others were convinced. A later example is the 1964 discovery of quarks, which are never seen in isolation. Murray Gell-Mann got the 1969 Nobel Prize for it, but the official citation did not mention quarks, and the introduction to his Nobel lecture only mentioned the great heuristic value of quarks. His lecture said that existence was immaterial, and that "quark is just a notion so far. It is a useful notion, but actual quarks may not exist at all."  His original paper only called it a "schematic model". Today quarks are commonly considered elementary particles.
Gell-Mann and the Nobel committee were being positivist, because without direct evidence for quarks, it is not necessary to believe that a quark is any more than a useful heuristic notion. Anti-positivists would say that, in retrospect, Gell-Mann was being unduly cautious.
It is not necessary to believe quarks are real particles, as there is barely any benefit to thinking of light as photons. Planck's original 1900 idea was that light was absorbed and emitted as discrete quanta, and that idea is still good today. You have to face some tricky paradoxes if you think of light always being particles.
A more recent example of positivism is the 2011 Nobel Prize for the discovery of the dark energy that permeates the universe. The recipients and most other astrophysicists refuse to say what it is, except that the phrase is a shorthand for the supernovae evidence for the accelerating expansion of the universe. When asked to speculate, one of them says "Reality is the set of ideas that predicts what you see."  Again, a positivist view prevails.
It is hard to find examples where positivist views have held back scientific progress, or where anti-positivist views have enabled progress. Weinberg claims to have a couple of examples,  such as J.J. Thomson getting credit (and the 1906 Nobel prize) for the discovery of the electron, even though a positivist did better work. Supposedly Thomson's non-positivism allowed him to see the electron as a true particle, and investigate its particle-like properties. Of course we now know that the electron is not truly a particle, and his son later got the Nobel prize for showing that it was a wave. Or a wave-like particle, or a particle-like wave, or a quantized electric field, or a renormalized bare charge, or whatever you want to call it. The positivist would say that the electron is defined by its empirical properties and our well-accepted quantum mechanical theories, and words like particle and wave are just verbal shortcuts for describing what we know about electrons.
Weinberg's larger point is that modern philosophy has become irrelevant and possibly detrimental to science. I would trace the problem to the rejection of logical positivism. That is when philosophers decided that they were no longer interested in truth, as understood by mathematicians and scientists.
Criticism of positivism also comes from the followers of T.S. Kuhn's paradigm shift theory. He denied that science was making progress towards truth, and portrayed the great scientific revolutions as irrational leaps to a theory with no measurable advantages. His favorite example was the /Copernican revolution, where a 1543 belief in the Earth's revolution around the Sun had no empirical advantages. 
Kuhn was influenced by the anti-positivist Michael Polanyi, who also drew anti-positivist lessons from Copernicus, and who claimed that relativity was discovered from pure speculation, intuition, and gedanken (thought) experiment, rather than from empiricism or real experiment. 
Polanyi's argument is as absurd as it sounds. The early papers on relativity had many startling consequences, including length contraction, time distortion, and mass increase, and every paper on these developments described them as consequences of the Michelson-Morley experiment. So do most of the textbooks, both now and a century ago. That was the crucial experiment. The sole support for Polanyi was Einstein's reluctance to credit others, but even some of Einstein's papers described the experiment as being crucial to relativity. Also, the prediction of mass increase with velocity was tested before Einstein wrote his first paper on the subject.
A positivist is likely to agree that the Ptolemy and Copernicus systems had little empirical difference. Indeed, general relativity teaches that geocentric and heliocentric coordinates are equally valid, as the physical equations can be written in any choice of coordinates.
Kuhn's paradigm shift theory really only applies to ancient debates over unobservables. XX century advances like relativity and quantum mechanics do not match his description at all, as they offered measurable advantages that a rational objective observer could appreciate.
Positivism has also become unpopular among some modern theoretical physicists who promote string theory, multiverses, and other ideas with no empirical support. A recent Nature article explains how "attempts to exempt speculative theories of the Universe from experimental verification undermine science". 
Logical positivist ideas have succeeded
Relativity and quantum mechanics are regarded as great theoretical advances in physics, and positivist ideas were essential in both. Relativity discards preconceptions about space, time, and mass, and replaced them with ideas more closely connected with what is measurable, and then relied heavily on experiment. The aether became a metaphysical idea whose existence was meaningless unless some way of measuring it were found.
Quantum mechanics, in the Copenhagen interpretation, made the observables the heart of the theory and openly declared the non-empirical questions to be meaningless. There is a long history of physicists and philosophers being unhappy with the positivism of this, and endorsing other interpretations that are supposedly more realist. Einstein famously denounced quantum mechanics, and others have also sought theories with hidden variables, pilot waves, and parallel universes. What they have in common is to assert the reality of unobservables, and to fail to provide an empirical test. No good has come from any of this anti-positivist work.
Nearly all good work in quantum mechanics follows the R.P. Feynman (misattributed) dictum "Shut up and calculate"  as well as "Unperformed experiments have no results." These slogans represent the positivist view that science is all about explaining and predicting what is observable, and avoiding speculation about what is not. The famous quantum paradoxes are all derived from anti-positivist and unnecessarily realist interpretations. 
Feynman did explain the virtues in having multiple theories to explain the same phenomenon, and those theories might have unobservable details. If those theories disagree about unobservables and agree about observable consequences, then they give a richer view of nature to the positivist, and not a disturbing contradiction. 
The verified theories are not falsified
Hypotheses are falsified all the time, and the possibility of falsification is an important way to analyze assertions of any kind. But well-accepted theories are almost never falsified. There are vast bodies of knowledge that the logical positivist can accept as verified. I consider some examples.
Euclid was not falsified by non-Euclidean geometry. Different postulates give different geometries. That is how geometry works.
The Ptolemy Almagest was a masterpiece of positivist scientific reasoning. It modeled the sky, as seen from Earth, and has not been proven wrong. His model was about as good as could be expected from the available data. His more speculative Planetary Hypotheses and his astrology treatise have not held up so well, but the Almagest was a scientific authority for a millennium.
The Almagest was geocentric in the same way that a modern personal computer planetarium program is geometric. It showed you what you see without modeling unobservables like distance from the Earth. Ptolemy also published a geography treatise. Complaining that his sky was geocentrist is like complaining that his Earth maps were flat. It should be obvious that he map is not the same as the territory.
Newton's theory of gravity has also held up remarkably well, and it still used today. If he were still alive, I think that he would be very surprised that anyone would argue that his theory was proved wrong. Instead, he might say that his theory turned out to be several orders of magnitude more accurate than he expected. After 300 years, there was a barely detectable anomaly in Mercury's orbit that was explained by relativity.
You might argue that the gravitational force is described differently in general relativity, but Newton would probably say that he was always unhappy with the action-at-a-distance aspect of his theory, and that relativity merely fills in a missing detail, thereby allowing greater accuracy under extreme conditions that he did not even consider. As a methodology for estimating trajectories and orbits, Newton's theory is as good as it ever was.
Concepts such as the aether and the epicycle are often said to have been proved wrong, but it is more accurate to say that they were refined with additional knowledge, just as the ancient concept of atoms has been refined. The main difference is that we have retained the word atom, and other terms have been replaced.
Even Aristotle's physics holds up pretty well. There is a widespread believe that he said stupid things like heavier objects falling faster than light ones, and that nobody dared question his treatise by doing the experiment. But he never said that, and there was no great reluctance to improve on his work. 
And modern science holds up even better. Sometimes scientists are wrong, such as those who denied continental drift, and there are lots of scientific controversies. But once something gets figured out and widely accepted, it is usually right. I do not agree with the negativists who say that all theories are either wrong, or not yet proven wrong.
Logical positivism gives a coherent view of modern math and science. It more closely resembles the views of modern successful mathematicians and scientists than the alternatives that philosophers espouse. In short, scientists believe that they are finding truth, and philosophers deny that it is possible.
(This essay was written for a philosophy site, but it has restricted to opinions from academic philosophers, and then shut down when that flopped.)
 A scientist argues that even math is empirical in this article: Defending scientism: mathematics is a part of science, Coel Hellier, http://scientiasalon.wordpress.com/2014/08/21/defending-scientism-mathematics-is-a-part-of-science/
 Metamath Proof Explorer Home Page, http://us.metamath.org/mpegif/mmset.html
 Logicism, http://www.encyclopediaofmath.org/index.php/Logicism
 APA 2014-2: Against causal reductionism. http://scientiasalon.wordpress.com/2014/12/31/apa-2014-2-against-causal-reductionism/
 Karl Popper attacked positivism, and may be considered a negativist. http://plato.stanford.edu/entries/popper/
 Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking, William H. Cropper, p.415.
 Brian P. Schmidt, 2014 lecture, podcast
 Dreams of a Final Theory, Steven Weinberg, http://www.phys.washington.edu/users/vladi/phys216/Weinberg_Against_philosophy.doc
 Paradigm shift, http://en.wikipedia.org/wiki/Paradigm_shift
 Michael Polanyi and the History of Science, http://www.missouriwestern.edu/orgs/polanyi/tad%20web%20archive/tad19-1/tad19-1-fnl-pg16-30-pdf.pdf
 Scientific method: Defend the integrity of physics, George Ellis & Joe Silk, http://www.nature.com/news/scientific-method-defend-the-integrity-of-physics-1.16535
 The Feynman dictum is actually due to Mermin. http://en.wikipedia.org/wiki/David_Mermin
 Quantum Theory Needs No ‘Interpretation’, Christopher A. Fuchs and Asher Peres, http://scitation.aip.org/content/aip/magazine/physicstoday/article/53/3/10.1063/1.883004
 The Character of Physical Law, Richard Feynman, http://en.wikipedia.org/wiki/The_Character_of_Physical_Law
 Aristotle's physics, Carlo Rovelli, http://philsci-archive.pitt.edu/10964/