Are there truths that can only be known through spiritual means? The motivation for this question basically comes from reading the discussion around the question Is it necessary to Biblically support arguments within discussions about Christianity? and a question at Philosophy Stack Exchange, Kierkegaard and the Tripartite Theory of Man in which the bipartite view (only mind and body) of man is distinguished from the tripartite view (spirit, soul and body).

In science, we know things through empirical methods, and in logic/mathematics, we know things through logical proofs. We use combinations of these methods to know various facts in daily life. Yet 1 Corinthians 2:14 says that "...the natural man receiveth not the things of the Spirit of God..." and uses the phrase "spiritually discerned" in reference to the things of the Spirit of God. Now in the book "The Spiritual Man" by Watchman Nee, which takes the tripartite view, there is reference made to "The Soul's Faculty of Intellect or Mind" (in Volume 1, Part 1, Chapter 2), hence apparently making a connection between "mind" and "soul". 

Hence all this leads to the obvious question: Are there true statements that inherently are unknowable by empirical means or logic and simply require "spiritual discernment"?

Perhaps the second question is: What is a man to do then?

Of course, I realize that this whole tripartite view might not even be widely accepted in the first place, nor might the book I mentioned be well-known. I myself do not claim expertise in all these matters. I'd just be happy to hear views and generate discussion. Thanks!
The short answer to your question is yes.
Matthew 7:28-29

28 And it came to pass, when Jesus had ended these sayings, the people were astonished at his doctrine:
29 For he taught them as one having authority, and not as the scribes.

What do you suppose it means, &quot;as one having authority, and not as the scribes&quot;? (Think how the scribes learned what they knew of the law.)
Galatians 1:12

12 For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.

Cool how Paul learned the gospel, isn't it? The disciples describe similar experiences thus:
Luke 24:32

And they said one to another, Did not our heart burn within us, while he talked with us by the way, and while he opened to us the scriptures?

The burning of the bosom is a sensation described multiple times in scripture as a possible way to know/recognize spiritual truth.
Paul describes the process of revelation by spiritual means to know God:
Ephesians 1:17-18

17 That the God of our Lord Jesus Christ, the Father of glory, may give unto you the spirit of wisdom and revelation in the knowledge of him:
18 The eyes of your understanding being enlightened; that ye may know what is the hope of his calling, and what the riches of the glory of his inheritance in the saints,

Clearly, it is by spiritual means that a personal testimony is gained.
Jesus directly told the disciples that the Holy Ghost will teach them/us all things:
John 14:26

26 But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.

Also, Romans is instructive:
Romans 8:16

16 The Spirit itself beareth witness with our spirit, that we are the children of God:

How is this spiritual truth learned? Spirit to spirit. I believe this is the only way to gain a true understanding and/or testimony of certain doctrines.
So what are we to do?
Jeremiah 29:13

12 Then shall ye call upon me, and ye shall go and pray unto me, and I will hearken unto you.
13 And ye shall seek me, and find me, when ye shall search for me with all your heart.

This promise is perhaps one of the most oft-repeated promises in all of holy scripture (for example, see Luke 11:9 or Matthew 7:7), and I think many, if not all, Christian faiths abide by this.