The director Spencer Gordon Bennet later noted that the film was influenced by the news of the voyage of Nautilus sub under the North Pole but that doesn't begin to account for this strange yet simple sci fi flick with political - or, more accurately, anti-political - overtones.
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Between Dan's conversations with other buddies, and the awkward voiceover narrator, we learn that the son is peacenik: he has opposed all war and apparently calls for negotiation with the Soviets (although that enemy is not mentioned explicitly--as it often is not in cold-war-era sci fi when other alien enemies stand in as analogies). The career of the war-hero father, we learn, has been cut short by the PR fiasco associated with the son's radical views. This is bad, bad, bad, and Dan cannot understand (thus neither can we) why a person with such views would be on this voyage. There are plenty of 50s-era sci fi films in which such a character - almost always dark-haired and swarthy - would turn out to be a "alien symp," a dupe who does the bidding of the little green men (in this case a furry tentacled super-intelligent cyclops who travels in an organic flying saucer) who want to take over the earth. But not here. There are a few moments when the camera focuses on the Brett Halsey's face, just when we are being asked to wonder who the real enemy is. But this never adds up to anything. In a way that's a relief, but what, then, is the purpose of this character?
Halsey's Dr. Neilson argues ideology with Dan ("We must give peace a chance"; "You fool, these enemies only know force"), but then the young dove reminds the captain that someone of his views can still be strong and principled and brave. Dan doesn't believe it, but when Dr. Neilson and Dan are the two mainly responsible for holding off the cyclops-like creature from outerspace it becomes clear that bravery trumps ideology any day. The evil creature is destroyed by an ICBM missile launched from the submarine and the world is saved. (It's a darned good thing America permitted itself to arm its subs with ICBM's. These are long-range missles and the cyclops was 100 feet away, but never mind.)
So in the final scene we're back in the U.S. of A., and Neilson and Dan are leaving the ship together. Shake hands; one last quick conversation. They like each other now, despite their opposing world views (as it were). Dan makes a wisecrack about having left his "little black book" on the alien's flying saucer, and Neilson very much appreciates this tragedy, so he's come a long way over to Dan's side. Maybe both of them will have blondes awaiting them ashore.
A conservative sci fi film would, as I say, have found Dr. Neilson guilty (the doctorate is almost always a bad sign) of sympathy or even collaboration with the enemy Other, or have him killed by said enemy after naively trying to deal with it. Liberal-centrist cold-war sci fi almost always enables us to see a Neilson type as redeemable - able to set his political ideas aside and see the value of use of force and the true beautiful humanity in each and every hawk (such as Don here).
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The movie doesn't deal with political difference so much as sets it aside in the natural course of staving off real enemies. The sons are not our real enemies; they've deluded themselves into thinking they disagree. Political difference is not relevant in a world being attacked by the Ultimate Enemy. And finally, natural commerce--to restore which was the reason for the rescue mission--in this new enterprise zone can bustle once again: that presumably profitable traffic going back and forth under the North Pole. All is well.