After a quick scan of Orthodoxy I have no idea what Chesterton is talking about, but I see two possible reasons why a Christian would criticise "pure rationalism":
1 - In the realm of beliefs/knowledge, empiricism has limits. Technically it is correct to say there is a limit to what we can know with our senses (pure rationalism). But we can't know what we don't know, so there is no way of debating what is beyond our senses, hence debate is futile.
2 - In the realm of values, "pure rationalism" is probably a misnomer for the culture forces unleashed by the Enlightenment (liberalism, individuality, permissiveness, cultural diversity). If so, Chesterton would have been wiser to direct his criticism at those values instead of rationalism because reason/rationalism are value-free data built from our senses. Values comes from emotion, not reason. Indeed, enlightened thinkers could subscribe to a traditional Christian-like culture.
1 - In the realm of beliefs/knowledge, empiricism has limits. Technically it is correct to say there is a limit to what we can know with our senses (pure rationalism). But we can't know what we don't know, so there is no way of debating what is beyond our senses, hence debate is futile.
2 - In the realm of values, "pure rationalism" is probably a misnomer for the culture forces unleashed by the Enlightenment (liberalism, individuality, permissiveness, cultural diversity). If so, Chesterton would have been wiser to direct his criticism at those values instead of rationalism because reason/rationalism are value-free data built from our senses. Values comes from emotion, not reason. Indeed, enlightened thinkers could subscribe to a traditional Christian-like culture.